RFC Review

Radio Science, Volume ???, Number , Pages 1–21,

Refractivity Estimation from Sea Clutter: An invited

ReviewAli Karimian,1 Caglar Yardim,1 Peter Gerstoft,1 William S. Hodgkiss,1

Amalia E. Barrios2

Non-standard radio wave propagation in the atmosphere is caused by anomalouschanges of the atmospheric refractivity index. In recent years, refractivity from clutter(RFC) has been an active field of research to complement traditional ways of measuringthe refractivity profile in maritime environments which rely on direct sensing of theenvironmental parameters. Higher temporal and spatial resolution of the refractivityprofile, together with a lower cost and convenience of operations have been the promisingfactors that brought RFC under consideration. Presented is an overview of the basicconcepts, research and achievements in the field of RFC. Topics that require moreattention in future studies also are discussed.

1. Introduction

Refractivity From Clutter (RFC) techniques es-timate the lower atmospheric refractivity structuresurrounding a radar using its sea surface reflectedclutter signal. The knowledge of the refractivitystructure enables radar operators to compensate fornon-standard atmospheric effects, or at least beaware of the radar limitations in specific locations.In the last decade, there has been interest in esti-mation of the environmental refractivity profile us-ing the radar backscattered signals. RFC can be de-scribed as a fusion of two disciplines [Rogers et al.,2000; Gerstoft et al., 2003b; Vasudevan et al., 2007]:numerical methods for efficient electromagnetic wavepropagation modeling and estimation theory.

Variations in the vertical refractivity profile can re-sult in entrapment of the electromagnetic waves, cre-ating lower atmospheric ducts. Ocean ducts are com-mon phenomena that result in significant variationsin the maximum operational radar range, creation ofradar fades where the radar performance is reduced,and increased sea clutter [Skolnik , 2008]. Therefore,they greatly alter the target detection performance

1Marine Physical Laboratory, Scripps Inst. ofOceanography, University of California, San Diego,California, USA.

2Atmospheric Propagation Branch, Space and NavalWarfare Systems Center, San Diego, California, USA.

Copyright 2012 by the American Geophysical Union.0048-6604/12/$11.00

at low altitudes [Anderson, 1995], and result in sig-nificant height error for 3–D radars.

RFC techniques find the profile associated withthe best modeled clutter match to the observed clut-ter power. RFC has the advantage of temporal andspatial tracking of the refractivity profile in a dynam-ically changing environment.

Atmospheric pressure, temperature and humidityaffect the refractivity structure, and thus affect theradar propagation conditions. The vertical gradientof the refractivity profile determines the curvatureof radar rays [Doviak and Zrnic, 1993]. Therefore,radar returns can be used to infer the gradient of re-fractivity structure near the ground [Park and Fabry ,2011].

Atmospheric ducts are more common in hot andhumid regions of the world. The Persian Gulf, theMediterranean and California coasts are examples ofsuch regions with common formation of a ductinglayer above the sea surface [Yardim et al., 2009]. Sur-face based ducts appear on an annual average almost25% of the time off the coast of South California and50% in the Persian Gulf [Patterson, 1992]. Whilesurface-based ducts appear less common than evap-oration ducts, their effect is more prominent on theradar return [Skolnik , 2008]. They often manifestthemselves in a radar plan position indicator (PPI)as clutter rings, see Fig. 1d, or height errors in 3–Dradars. The height error is due to the trapping of thelowest elevation beams near the surface instead of re-fracting upward as would be expected in a standardatmosphere.

1

2 KARIMIAN ET AL.: RFC REVIEW

-250!

-200!

-150!

-100!

-50!

0!

50!

100!

150!

200!

250!

-250!

-200!

-150!

-100!

-50!

0!

50!

100!

150!

200!

250!

km

-200 -100 0 100 200! -200 -100 0 100 200!

km km

(a)! (b)!

(c)! (d)!

(dB)! (dB)!

(dB)!

Figure 1. Propagation diagram of a (a) weak evaporation duct, (b) surface-based duct (high inten-sity: bright). Radar PPI screen showing clutter map (dB) during the 1998 SPANDAR experimentresulting from a (c) weak evaporation duct, (d) surface-based duct.

Fig. 1b shows that a surface-based duct increasesthe radar range significantly inside the duct with re-spect to a weak evaporation duct (close to the stan-dard atmosphere) by trapping the radar waves justabove the ocean surface. Note that the electromag-netic energy is trapped inside the strong surface-based duct which results in an increase in the in-teraction of the electromagnetic waves with the seasurface. Fig. 1(c,d) demonstrates the effect of atmo-spheric ducts on the radar clutter. The strong duct-ing case has distinct clutter rings around the radar.This complex clutter structure enables RFC to esti-mate the atmospheric conditions from the radar re-turns.

Efforts by Reilly and Dockery [1990] and Pappertet al. [1992] to calculate sea reflections in ductingconditions inspired researchers to find the environ-mental refractivity profile from radar measurements,as opposed to the traditional way of using bulk sensormeasurements. The atmospheric refractivity profileis often measured by direct sensing of the environ-ment. Rocketsondes and radiosondes typically areused for sampling of the atmospheric boundary layer[Rowland et al., 1996], although they have limita-

tions regarding mechanical issues and surface condi-tions [Helvey , 1983; Mentes and Kaymaz , 2007]. Forcharacterization of the surface layer, “bulk” param-eters such as pressure, air and sea surface temper-ature, humidity, and wind speed are measured at asingle height, usually with sensors placed on a buoyor platform on the sea surface. These in-situ mea-surements are then used as inputs to thermodynamic“bulk” models to estimate the near-surface verticalrefractivity profile using Monin–Obukhov similaritytheory [Jeske, 1973; Fairall et al., 1996; Fredericksonet al., 2000b].

Initial remote sensing studies in the radar[Richter , 1995; Rogers, 1997] and climatology [Haackand Burk , 2001] communities have been directed to-ward a better estimation of the refractivity profilein the lower atmosphere, less than 500 m above thesea surface. Hitney [1992] demonstrated the ca-pability to assess the base height of the trappinglayer from measurements of UHF signal strengths.Anderson [1994] inferred vertical refractivity of thelower atmosphere based on ground-based measure-ments of global positioning system (GPS) signals,followed by Lowry et al. [2002]; Lin et al. [2011].

KARIMIAN ET AL.: RFC REVIEW 3

Boyer et al. [1996] estimated refractivity from radiomeasurements with diversity in frequency and height.Rogers [1997] used VHF/ UHF measurements fromthe VOCAR 1993 experiment to invert for a three pa-rameter (base height, M deficit and duct thickness)surface duct model. Krolik and Tabrikian [1997] useda maximum a posteriori (MAP) approach for inver-sions. They modeled the environment with a threeelement vector: two elements to describe the verticalstructure and one to describe the range dependencyof the profile. They later combined prior statisticsof refractivity with point-to-point microwave propa-gation measurements to infer refractivity [Tabrikianand Krolik , 1999].

Von Engeln et al. [2003] used low earth orbitGPS satellites to analyze the occurrence frequencyand variation of land and sea ducts on a globalscale, during a 10 day period in May 2001. LIDAR[Wandinger , 2005; Willitsford and Philbrick , 2005]has also been used to measure the vertical refractiv-ity profile. However, its performance is limited bythe background noise (e.g. clouds) [Vasudevan et al.,2007].

Weather radars and refractivity retrieval algo-rithms have been used to estimate moisture fieldswith high temporal and spatial resolution [Fabryet al., 1997; Weckwerth et al., 2005; Roberts et al.,2008] with application in understanding thunder-storm initiation [Wilson and Roberts, 2006; Waki-moto and Murphy , 2009].

RFC techniques use the radar return signals to es-timate the ambient environment refractivity profile.There has been strong correlation between the re-trieved refractivity profile using an S-band radar andin-situ measurements by instrumented aircrafts or ra-diosondes [Rogers et al., 2000; Gerstoft et al., 2003b;Weckwerth et al., 2005]. RFC techniques make track-ing of spatial and temporal changes in the environ-ment possible [Vasudevan et al., 2007; Yardim et al.,2008; Douvenot et al., 2010]. RFC inversions of theenvironmental profile have been reported at frequen-cies as low as VHF [Rogers, 1997], and as high as5.6 GHz [Barrios, 2004].

The development of RFC initially was inspired bythe use of inverse methods in ocean acoustics whichalso is based on propagating signals in a waveguide.For a review of numerical modeling of the oceanwaveguide see [Jensen et al., 2011]. For an introduc-tion to the ocean acoustic inverse problem see [Dosso

and Dettmer , 2011] and for sequential inverse meth-ods in ocean acoustics see [Yardim et al., 2011].

The remainder of this paper is organized as fol-lows: Section 2 introduces the marine ducts and theirsimplified mathematical models. Section 3 summa-rizes the clutter models used in previous studies andwave propagation approximations that model radiowave propagation efficiently. Section 4 summarizesthe RFC research and inversion methods that havebeen used to infer the environmental refractivity pa-rameters. Section 5 discusses the shortcomings of thecurrent research and areas that require more atten-tion in the future.

2. Marine ducts

One of the first reports of abnormal performanceof radar systems in maritime environments was dur-ing World War II where British radars on the north-west coast of India commonly observed the coast ofthe Arabian peninsula 2700 km apart under monsoonconditions [Kerr , 1951]. Marine ducts are the re-sult of heat transfer, moisture and the momentum ofchanges in the atmosphere [Gossard , 1981] and en-tail three general classes: evaporation, surface-basedand elevated ducts.

(a)!

hd

(b)!

h1

h2

m1

m2

(d)!

h1

h2

m1

m2

(c)!

hdm1

h2h1

m2

M-units! M-units!

Figure 2. Parameters of simplified duct geometries:(a) evaporation duct, (b) surface-based duct, (c) surface-based duct with an evaporation layer, and (d) elevatedduct.


These ducts are characterized by a range andheight dependent environmental refractivity index.Although a refractivity profile has a complex struc-ture in nature, it can be approximated by a bilin-ear or trilinear function for surface-based ducts andby an exponential function for evaporation ducts inmodeling wave propagation [Dougherty and Hart ,1979; Rogers, 1996; Gerstoft et al., 2003b].

The simplified atmospheric duct geometries usedin most RFC works are shown in Fig. 2. The mod-ified refractive index M is defined as the part permillion deviation of the refractive index from that ofa vaccum:

M(z) · 10−6 = n(z)− 1 + z/re, (1)

which maps the refractivity index n at height z toa flattened earth approximation with earth radiusre = 6370 km. The advantage of working with themodified refractive index is to transform a spher-ical propagation problem into a planar one. Thistransformation maps a spherically stratified mediumover a spherical earth to a planar stratified mediumabove a flat earth. This transformation results in lessthan 1% error for ranges of less than re/3, indepen-dent of the wavelength [Pekeris, 1946]. However, thistransformation to compute the height–gain functionbreaks down in centimeter wavelengths and elevationof more than 300 m. The error gets worse with in-creasing frequency [Pekeris, 1946].

2.1. Evaporation ducts

Existence of evaporation ducts was first suggestedby Katzin et al. [1947]. Because of the difficultiesin directly measuring the evaporation duct, variousbulk models have been used to estimate the near-surface refractivity profile for several decades [Jeske,1973; Liu et al., 1979; Paulus, 1985; Babin et al.,1997]. An evaporation duct model that assumes hor-izontally varying meteorological conditions has beensuggested by Greenaert [2007]. Examples of suchconditions are reported to frequently happen in thePersian Gulf [Brooks et al., 1999]. One of the morewidely accepted high fidelity evaporation duct mod-els which has been used in various evaporation ductresearch studies is the model developed by the NavalPostgraduate School [Frederickson et al., 2000a]. A4-parameter model for range independent evapora-tion ducts that controls the duct height, M–deficitand slope has been suggested by Zhang et al. [2011b].

The Paulus–Jeske (PJ) evaporation duct model ismore commonly used operationally due to its empir-

ical correction for spuriously stable conditions. ThePJ model is based on the air and sea surface tem-peratures, relative humidity, wind speed with sensorheights at 6 m and the assumption of a constant sur-face atmospheric pressure [Jeske, 1973; Paulus, 1985;Babin et al., 1997]. For the neutral evaporation duct,where the empirical stability functions approach aconstant, the PJ model is simplified to [Rogers et al.,2000]:

M(z) = M0 + c0(z − hd lnz + z0

z0), (2)

in which M0 is the base refractivity, c0 = 0.13 M-unit/m corresponding to the neutral refractivity pro-file as described by Paulus [1990], z0 is the rough-ness factor taken as 1.5× 10−4 m, and hd is the ductheight. The exact choice of M0 (usually taken inthe interval [310–360] M-units/m) does not affect thepropagation pattern since it is the derivative of Mthat dictates wave propagation in the medium [Hit-ney , 1994; Gerstoft et al., 2000].The assumption ofneutral stability implies that the air and sea-surfacetemperature difference is nearly zero, and wind speedis no longer required. It was found in [Rogers andPaulus, 1996] that propagation estimates based ona neutral-stability bulk model performed well rela-tive to other more sophisticated bulk models for themeasurement sets under consideration. This is animportant point as all RFC-estimated evaporationduct heights, and subsequently evaporation duct pro-files given in (2), are based on neutral conditions.

2.2. Surface-based ducts

Surface ducts typically are due to the advection ofwarm and dry coastal air to the sea. The trilinearapproximation of the M-profile, as shown in Fig. 2b,is represented by:

M(z) = M0 +

m1z z ≤ h1

m1h1 +m2(z − h1) h1 ≤ z ≤ h2

m1h1 +m2(h2 − h1) h2 ≤ z+m3(z − h1 − h2)

(3)where m3 = 0.118 M-units/m, consistent with themean over the United States. Since profiles are up-ward refracting, clutter power is not very sensitive tom3 [Gerstoft et al., 2003b].

A surface duct, schematically shown in Fig. 2c, hasalso been used by Gerstoft et al. [2003b]; Rogers et al.[2005], which includes an evaporation duct layer be-


neath the trapping layer:

M(z) = M0+

M1 + c0

(z − hd ln z+z0

z0

)z ≤ zd

m1z zd ≤ z ≤ h1

m1h1 −Mdz−h1

zthickh1 ≤ z ≤ h2

m1h1 −Md +m3(z − h2) h2 ≤ z(4)

where c0 = 0.13, m1 is the slope in the mixed layer,m3 = 0.118 M-units/m, h1 is the trapping layer baseheight, and zd is the evaporation duct layer heightdetermined by:

zd =

{ hd

1−m1/c00 < 1

1−m1/c0< 2

2hd Otherwise(5)

subject to zd < h1. h1 = 0 simplifies (4) to abilinear profile and h2 = 0 implies standard at-mosphere. zthick is the thickness of the inversionlayer, and h2 = h1 + zthick. M1 is determined byM1 = c0hd ln zd+z0

z0+ zd(m1 − c0), and Md is the M-

deficit of the inversion layer. Gerstoft et al. [2003b]used an 11 parameter model for the environmentalrefractivity profile: five paremters for the verticalstructure as in (4), and six to model the range varia-tions of the profile. They assumed that the trappinglayer height h2 is range dependent and used principlecomponents of h2 as a Markov process with respectto range.

Most of the RFC studies including [Gerstoft et al.,2004; Yardim et al., 2007; Vasudevan et al., 2007]have used a four parameter surface based duct.

However, the frequency range of the validity ofa trilinear approximation to the surface duct re-fractivity structure is arguable. As Fig. 3 demon-strates, the trilinear approximation to complex re-fractivity profile structures gets worse for modelingwave propagation at higher frequencies. Propaga-tion loss and clutter power of a measured profileand its trilinear approximation are shown in this fig-ure. The profile is from the SPANDAR 1998 dataset(Run 07, range 50 km) measured by an instrumentedhelicopter along the 150◦ azimuth shown in Fig. 1[Rogers et al., 2000]. Panel (a) shows the trilinearapproximation obtained by minimizing the l2 normof the difference of the approximated and real pro-files given that the slope of the third line is fixed andequal to 0.12 M-units/m. Panel (b) shows the prop-agation loss of the measured profile with antennaheight of 25 m, frequency of 3 GHz, beamwidth of0.4◦ and wind speed of 5 m/s. The propagation lossis obtained from the Advanced Propagation Model

[Barrios, 2002] which uses a parabolic equation code[Barrios, 1994]. The clutter power is obtained from amultiple angle clutter model [Karimian et al., 2011b].Panels (c) and (d) show that the error of the trilinearapproximation for a complicated structure increaseswith frequency. Here, the average absolute error ofthe propagation loss inside the duct increases from4.9 dB at 3 GHz to 6.7 dB at 10 GHz. The absolutevalue of the clutter power difference due to the mea-sured refractivity profile and its trilinear approxima-tion increases from the average of 8.2 dB at 3 GHzto 13.3 dB at 10 GHz. However, experimental mea-sured profiles show that the trilinear approximationis sufficient for most of the surface-based ducts, es-pecially when propagation is to be modeled at 3 GHzand lower frequencies [Gerstoft et al., 2003b].

A wavelet representation of the conductivity pro-file was suggested in the similar inverse scatteringproblems arising in geophysical prospecting [Millerand Willsky , 1996a, b]. Generalized Karhunen–Loeve transform [Hua and Liu, 1998] was used byKraut et al. [2004] to find the the tropospheric re-fractivity basis vectors of VOCAR 1993 profiles mea-sured off the coast of California. Both of these ap-proaches are capable of representing environmentalprofiles in more detail with additional complexity ininversions.

2.3. Elevated ducts

Elevated ducts, schematically shown in Fig. 2d,are unstable atmospheric conditions that are primar-ily observed over the land but may also be formedacross the seashore when cool air flows over a warmersea [Guinard et al., 1964; Gossard , 1981; Kukushkin,2004]. The effects from these types of ducts arenot visible on a radar screen since radar beams gettrapped in the elevated layer above the ocean level.Elevated ducts might be predicted from the natureof heat absorbing and radiating boundaries and thecloud cover [Gossard , 1981].

3. Electromagnetic theory and forwardmodeling

Given a refractivity structure m in a maritime en-vironment, the expected clutter power is obtained asa function of radar and environmental parameters.Assuming that electromagnetic waves hit the surfaceat a single grazing angle at range r, the received radar


300 310 320 3300

100

200

300(a)

Alt

itude

(m)

M!units

measured prof.

trilinear approx.

(b)

20 40 60 800

50

100

150

200

!20

0

20

Alt

itude

(m)

(c)

20 40 60 800

50

100

150

200

0

10

20

(d)

20 40 60 800

50

100

150

200

0 20 40 60 80

!30

!20

!10

0

(e)

Range (km)

clutt

er p

ow

er (

dB

)

0 20 40 60 80!30

!20

!10

0

(f)

Range (km)

(dB)!

Figure 3. (a) A measured profile from the 1998 SPANDAR and its trilinear approximation. (b)propagation loss (dB) of the measured profile at 3 GHz. Propagation loss difference of the measuredprofile and the trilinear approximation at (c) 3 GHz, (d) 10 GHz. Clutter power comparison of theprofile and its trilinear approximation at (e) 3 GHz, (f) 10 GHz.

power is [Dockery , 1990; Skolnik , 2008]:

Pr(r) =PtGAeσF

4(r,m)

(4π)2r4L, (6)

where Pt is the transmitter power, G the antennagain, Ae the antenna effective aperture, σ the ef-fective cross section of the scatterer, L the total as-sumed system losses, and F is the propagation factorat the sea surface. The pattern propagation factor Fis defined as the ratio of the magnitude of the electricfield at a given point under specified conditions to themagnitude of the electric field under free-space con-ditions [Kerr , 1951]: F (r) = |E(r)|

|Efs(r)| . F is a functionof range r and the refractivity structure m at eachlocation. The antenna effective aperture is obtainedas a function of the wavelength λ, Ae = λ2G

4π . Theclutter cross-section σ becomes σ = Acσ0 where σ0

is the clutter cross section per unit area and Ac isthe area of the radar cell [Skolnik , 2008]:

Ac = rθB(cτ/2) sec (θ(r,m)) , (7)

with θB the antenna pattern azimuthal beamwidth,c the propagation speed, τ the pulse width, and θ isthe grazing angle which is a function of range andthe environmental refractivity. From this point on,F (r,m) and θ(r,m) are shown as F and θ for sim-plicity. Thus, the clutter power at the range r isobtained as:

Pc(r) =PtG

2λ2θBcτσ0 sec(θ)F 4

2(4πr)3L. (8)

The propagation factor F is calculated by numer-ical solutions to the wave propagation problem (Sec-tion 3.1). The sea surface-reflectivity per unit areaσ0 is calculated from semi-empirical models that fitthe experimental measurements to a function of sys-tem parameters (Section 3.2).

The angle with which electromagnetic waves hitthe ocean surface θ varies with range. However, thedependence of the clutter model on grazing anglehas been neglected at far distances from the radar


in [Rogers et al., 2000, 2005; Gerstoft et al., 2003b;Kraut et al., 2004; Yardim et al., 2006, 2008; Vasude-van et al., 2007; Wang et al., 2009; Douvenot andFabbro, 2010; Zhao et al., 2011]. The sec(θ) termalso is a weak function of θ at low angles. Thus,normalization of the clutter power by the power atrange r0 yields the approximation:

Pc(r)

Pc(ro)'(ror

)3 F 4(r)

F 4(ro). (9)

Rogers et al. [2000] considered the dependency ofthe sea-surface reflectivity with grazing angle in anevaporation duct and concluded that σ0 ∝ θ0 giventhat the clutter cells are far enough from the radar.They also investigated the existence of a minimumwind speed under which radar return is not reliablefor duct height inversion. The minimum wind speed(usually less than is 2 m/s) depends on the radar pa-rameters and sensitivity.

To overcome the problem of uncertainty of σ0, geo-metrical ray tracing and rank correlation was used byBarrios [2004] for inversion of surface-based ducts.

The assumption that there is a single grazing an-gle θ at each range is not always valid, especiallyin strong surface-based ducts where multiple electo-magnetic waves with different angles hit the surfaceat each location. Karimian et al. [2011b] suggesteda clutter model that depends on all grazing anglesproportional to their relative powers:

Pc(r) =αc(r)F

4(r)∫θγ(θ)dθ

∫θ

σ0,GIT (θ) sec(θ)γ(θ)

F 4std(θ)

dθ,

(10)

where αc(r) = PtG2λ2θBcτ

2(4πr)3L includes all grazing an-gle independent terms, σ0,GIT is the sea surface re-flectivity from the GIT model (discussed in Section3.2), γ(θ) is the relative energy of incident wavefrontsat each grazing angle obtained from a curved wavebeamformer, and Fstd(θ) is the propagation factorof a standard atmosphere at a range with the samegrazing angle. An analysis of the performance of dif-ferent clutter models in RFC inversions is providedin [Karimian et al., 2011a].

3.1. Wave propagation modeling

From the early days of wave propagation mod-eling, a divergence arose due to the distinct differ-ences in applications emphasizing environmental ef-fects over terrain versus over the oceans. Due to theadvances in computer processing as well as innova-tive mathematical techniques for numerically inten-

sive problem solving, the most popular techniques forRadio Frequency (RF) propagation modeling haveconverged such that these same methods are wellsuited for both land and water propagation paths.Since the emphasis of this paper is on the estimationof refractive conditions over the ocean, this sectionwill describe only those RF propagation modelingtechniques and algorithms as they pertain to mod-eling anomalous propagation effects on over-waterpaths.

One of the first radiowave propagation models thattook into account the effects of both evaporationducts and surface-based ducts was based on the tech-niques described in [Kerr , 1951] and [Blake, 1980].The model determines the coherent sum of the di-rect and surface-reflected fields within the optical in-terference region, also accounting for divergence andnon-perfect reflection by use of a modified Fresnel re-flection coefficient [Hitney and Richter , 1976]. Mod-eling refractive effects is limited since within this re-gion, the use of an effective earth radius factor is em-ployed to account for non-standard conditions. Fordiffraction effects beyond the radio horizon, duct-ing effects are based on a single mode model wherean empirical fit to waveguide solutions are used tomodify Kerr’s standard diffraction method [Hitney ,1994].

For modeling of height-varying refractive condi-tions, waveguide models offer a much higher fidelitysolution and have been in use since the early 1900s[Budden, 1961]. Waveguide models employ normalmode theory and are well suited when refractive con-ditions do not change along the path. Due to thehigh computational requirements for mode searches,another caveat is that normal mode models are typ-ically used beyond the radio horizon where far fewermodes are needed for a solution [Pappert et al., 1992;Hitney , 1994].

One of the more popular techniques for RF prop-agation modeling is the parabolic equation (PE)method, also known as the paraxial approximationmethod. Originally used by Leontovich and Fock[1946], the PE method allows for propagation con-ditions to vary in both height and range. However,the PE method was not in practical use until Hardinand Tappert [1973] developed a technique called thesplit-step Fourier (SSF) method, initially applied tounderwater acoustic propagation. The SSF methodtook advantage of fast Fourier transforms that led toextremely efficient numerical solutions of the PE. Koet al. [1983]; Skura et al. [1990] modified the under-


water acoustic SSF PE to model radiowave propaga-tion in the troposphere. Since that time many im-provements and mathematical techniques have beenintroduced in the SSF PE algorithm for applicationsto RF propagation in the troposphere. For an ex-cellent treatise on the development of many of thesetechniques, the reader is referred to Levy [2000].

Due to its efficiency and accuracy the SSF PEalgorithm is now widely used in many radiowavepropagation models, including the model used hereto obtain results presented in this paper. A gen-eral description of the SSF PE algorithm is given inthe following, with more details provided on specificimplementation of the model used here. Applyingthe simple assumption of a slowly varying medium,Maxwells equations can be reduced to the scalar two-dimensional (Cartesian) elliptical Helmholtz equa-tion:

∂2ψ(x, z)

∂x2+∂2ψ(x, z)

∂z2+ k2

0n2ψ(x, z) = 0, (11)

where ψ(x, z) is a function of the electric or magneticfield, depending on the polarization of the radiatedfield; and n is the refractive index of the medium(implicitly also a function of x and z). The usualstarting point for the derivation of the PE is sub-stituting the function ψ(x, z) = ejk0xu(x, z) in (11),then factor the result into, respectively, forward andbackward pseudo-differential equations:

∂u(x, z)

∂x+ jk0

[1−

√1

k20

∂2

∂z2+ n2

]u(x, z) = 0, (12)

∂u(x, z)

∂x+ jk0

[1 +

√1

k20

∂2

∂z2+ n2

]u(x, z) = 0. (13)

This substitution effectively removes the rapid phasevariation in ψ, leaving u(x, z) a slowly varying func-tion in range. In most PE models used for long rangeradiowave tropospheric propagation, only the for-ward propagating term (12) is solved, and the back-ward propagating term is ignored.

Initial PE algorithms incorporated simple approx-imations to (12), resulting in the standard PE (SPE).The limitation with using the SPE is that it isa narrow-angle approximation and leads to largererrors when propagating at large angles, typicallygreater than 10◦ for microwave frequencies. Feit andFleck [1978] developed the wide-angle PE (WAPE)for propagation within optical fibers, by using an al-ternative approximation of the square-root operator.

Later, Thomson and Chapman [1983] quantified theerror associated with the use of various approxima-tions to the square-root operator, concluding that theWAPE propagator developed by Feit and Fleck wasa substantial improvement in reducing phase errorsat large propagation angles necessary for their workin underwater acoustic propagation. More recently,Kuttler [1999] analyzed the differences between theSPE and WAPE and offered yet a further improve-ment for the WAPE and wide-angle sources.

The Leontovich surface impedance boundary con-dition must then be applied to obtain a solution forthe WAPE:

∂u

∂z

∣∣∣∣z=0

+ αu∣∣z=0

= 0, (14)

where the complex α is given by

αh,v = jk0 sin θ

[1− Γh,v1 + Γh,v

]. (15)

Here, θ is the grazing angle of the radiated field at thesurface, Γ is the Fresnel reflection coefficient - alsodependent on the grazing angle, and the subscriptsh and v refer to horizontal and vertical polarizationrespectively. The discrete mixed Fourier transform(DMFT) formulation provided by Dockery and Kut-tler [1996] implements the impedance boundary con-dition and derives the new split-step solution entirelyin the discrete domain. The DMFT method has theadded advantage that it retains numerical efficiencydue to requiring only sine transforms. Further re-finement of the DMFT was presented by Kuttler andJanaswamy [2002] where they applied various differ-ence formulations for (14) to arrive at an improvedDMFT algorithm, reducing much of the numericalinstabilities associated with the quantity αh,v whenRe(αh,v) approaches zero.

The propagation model used for the results pre-sented in this paper implements the WAPE andthe DMFT algorithm as described in Kuttler [1999];Dockery and Kuttler [1996]; Kuttler and Janaswamy[2002] and is called the Advanced Propagation Model(APM). The handling of range-varying vertical re-fractive profiles is described in [Barrios, 1992] and ageneral description of the APM is provided in [Bar-rios, 2003].

Pertinent to the RFC methodology is the accuracyof the forward scattered field, which is subsequentlydependent on how αh,v is modeled. Typically, theboundary condition is modeled such that a constantimpedance is assumed within each range step, de-


350 4000

100

200

300

400

500

600

700

800

900

1000

M− units

Alti

tude

(m

)(a)

0 100 200 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Range (km)

(b)

Ang

le (d

eg)

Figure 4. (a) Refractivity profile of surface-based ductused for (b) determination of grazing angles by ray trace(solid) and final maximum angles (dashed lines) used forcomputing αh,v.

pendent on a single grazing angle associated withthe dominant mode of propagation for the specifiedrefractive environment. We apply the Kirchoff ap-proximation and model the sea surface boundary bydetermining an effective impedance described by areduction, ρ, to the smooth surface Fresnel reflec-tion coefficient, Γ0, based on the Miller-Brown-Vegh(MBV) model [Miller et al., 1984]:

Γh,v = ρΓ0h,v (16)

ρ = e−2(2πγ)2I0[2(2πγ)2

](17)

γ =hw sin θ

λ(18)

I0 is the modified Bessel function of the first kind,and hw is the rms wave height from the Phillips oceanwave spectrum [Phillips, 1985]:

hw = 0.0051v2w, (19)

where vw is the wind speed in m/s. Within APM, ρis approximated according to ITU reports [1990] bythe expression

ρ =1√

3.2χ− 2 +√

(3.2χ)2 − 7χ+ 9, (20)

χ = 8π2γ2. (21)

Next is to determine the grazing angle at each PErange step to compute the effective reflection coeffi-cient and subsequent impedance. Grazing angles at

the sea surface can easily be found using a geomet-ric ray trace based on small angle approximationsto Snell’s law [Dockery et al., 2007]. The caveat isthat for surface-based ducting conditions, there willbe multiple grazing angles within a given range in-terval/step, as shown in Fig. 4. Figure 4(a) showsthe refractivity profile of a 300 m surface-based duct,and the corresponding grazing angles are shown inFig. 4b. Notice that beyond the skip zone, at rangesbeyond 80 km, there are multiple grazing angles (i.e.,multiple modes) present within a given range inter-val. The challenge is determining the proper grazingangle associated with the dominant mode of prop-agation at a particular range. Geometric ray trac-ing techniques offer no further information, therefore,spectral estimation techniques have also been used[Dockery and Kuttler , 1996; Schmidt , 1986; Barrios,2003] in combination with geometric ray trace meth-ods to obtain the appropriate angle at a given rangeparticularly useful in complex environments wherethe propagation path is a combination of sea, land,and a range-dependent atmosphere.

Of course, one of the caveats of modeling theimpedance in this way is that for surface-based duct-ing environments it ignores the many, equally dom-inant, modes propagating within the duct at multi-ple grazing angles within a range step. The advan-tage of using the MBV method to modify the sur-face impedance is that it is easy to implement andfor the most part has been shown to perform verywell for range-independent evaporation duct environ-ments where the incident field can be described, toa very good approximation, by a single grazing an-gle beyond the interference region [Anderson, 1995;Rogers et al., 2000].

A more rigorous, albeit conventional, approachhas been provided by Janaswamy [2001] to modela non-constant impedance that directly takes intoaccount effects of the angle-dependent reflection co-efficient present at all grazing angles. However, inkeeping with the more numerically efficient SSF PEapproach, and considering the design toward op-erational applications, the maximum grazing angle(shown by the dashed line in Fig. 4b) is used in com-puting αh,v to model rough surface effects. This re-sults in maximum, or worst-case, clutter values andwill in general over-estimate sea clutter.

Finally, a recent approach to more accuratelymodel the various field strengths at the surface, andsubsequently, clutter power described by multiplegrazing angles, has been provided by Karimian et al.


[2011b] that takes all grazing angles and their rela-tive powers at each range-step into account.

For the RFC application, the propagation factor,F , in the clutter equation (8–10) is a function ofthe complex PE field and the range (note that rangeis shown by r in the clutter equations and by x inthis section, since Maxwell’s equations are solved inCartesian coordinates):

F = |u(x, zeff)|√x, (22)

where zeff is the effective scattering height, taken as0.6 times the mean wave height [Reilly and Dockery ,1990], or approximately 1 m above the ocean for mostsituations [Rogers et al., 2000; Barrios, 2003]. The-oretically, F should be computed from the incidentfield at the sea surface. However, PE approximationsyield the propagation factor due to the total fieldwhich is close to zero at the sea surface and high fre-quencies. Konstanzer et al. [2000] showed that theclutter power using the total field propagation fac-tor at the effective scattering height is proportionalto the clutter power using the incident propagationfactor.

3.2. Sea surface reflectivity models

Proper characterization of the quantity σ0F4 in

(8) is key to providing reasonable clutter predictionsto perform RFC. The difficulty is that the surface

(a) (b)

0.1 1 10−80

−70

−60

−50

−40

−30

−20

Surfa

ce re

flect

ivity

(dB

)

Grazing angle (deg)

(a)

0.1 1 10Grazing angle (deg)

(b)

GITTSCSITBAR

Grazing angle (deg)! Grazing angle (deg)!

Figure 5. Reflectivity vs. grazing angle for several seasurface reflectivity models at (a) 3 GHz, and (b) 9.3 GHz.

reflectivity is implicitly dependent on the forwardpropagation effects defined by F . They are inher-ently coupled yet these two quantities are commonlytreated separately to get an estimate of the returnclutter. Most sea surface reflectivity models, there-fore, are semi-empirical and are based on site-specificpropagation data, typically with no correspondingmeteorological measurements.

There are several semi-empirical models for the av-erage sea surface reflectivity per unit area that fit theexperimental sea clutter data to a function of radarfrequency, grazing angle, beam width, wind speed,radar look direction with respect to the wind, andpolarization. This quantity, represented by σ0, isalso referred to as the normalized radar reflectivity[Nathanson et al., 1991].

A hybrid model by Barton [1988] and the GeorgiaInstitute of Technology (GIT) model [Horst et al.,1978] are among the classic sea surface reflectiv-ity models for low grazing angles that are valid inthe S and X band frequencies. A comparison ofdifferent models is provided in [Reilly and Dock-ery , 1990]. GIT, Technology Services Corp. (TSC)[Fletcher , 1978], and Barton (BAR) reflectivity mod-els at 3 GHz are compared in Fig. 5a. A similar com-parison at 9.3 GHz with the additional Sittrop (SIT)[Sittrop, 1977] model is shown in Fig. 5b. Notice thatthe TSC, BAR, and SIT models show similar depen-dence of σ0 on grazing angle, whereas the GIT modelexhibits higher attenuation at lower grazing angles.Lower grazing angles imply the region near the radiohorizon subject to diffraction effects. The increasedattenuation shown by the GIT model as a functionof decreasing grazing angle is indicative of standarddiffraction effects, and it is for this reason the GITmodel has been more widely used. That is, the GITreflectivity can be assumed to be representative of σ0

under standard atmosphere conditions.Reilly and Dockery [1990] modified the GIT model

to consider ducting effects on the radar backscatterby dividing σ0 by the standard atmosphere propaga-tion factor and multiplying by the propagation factorof the desired conditions [Dockery , 1990].

Normalized mean sea backscattering coefficient σ0

for grazing angles of 0.1 to 60◦ and frequencies of 0.5to 35 GHz are tabulated by Nathanson et al. [1991]based on almost 60 experiments. A model to fitthe aforementioned dataset for grazing angles lessthan 10◦ and frequencies up to 35 GHz is providedby Gregers-Hansen and Mital [2009]. Modeling the


sea surface reflectivity suitable for RFC applicationsremains an active field of research.

Calculation of the grazing angle is the key to thecalculation of radar backscatter. A hybrid of raytracing and plane wave beamforming has been sug-gested in the works of Dockery and Kuttler [1996];Barrios [2003]; Dockery et al. [2007] to find the an-gle of arrival based on the propagation conditions.Karimian et al. [2011b] suggested a curved wavebeamformer that depends on the refractivity profileat each location.

4. Inverse problem framework

The radar clutter depends on the two way prop-agation loss from the transmitter to the range cell.The loss in turn depends on the environmental re-fractivity profile through which the wave is propa-gated. The expected clutter power of each candidateprofile is computed and an objective function Φ thatquantifies the difference between the observed, Po,and the simulated clutter power, Ps(m), is formed.Po and Ps are the vectors of clutter power over theradar range. The candidate profile that yields theminimum difference is declared as the best match.

m = argminm

Φ(Po,Ps(m)). (23)

The simulated clutter is a function of the propaga-tion factor F , as seen in (6). F in turn, is a functionof the environmental profile m. Using an l2 norm asthe objective function Φ yields:

Φ = ‖Po −Ps(m)‖2, (24)

which is also the negative log-likelihood function un-der the Gaussian noise assumption. Minimizing (23)over the refractivity profile m requires an efficientnumerical search for the optimum values.

There have been several approaches to estimatethe refractivity parameters from the observed clutterincluding: a matched–field processing approach to-ward inversion [Gerstoft et al., 2000], a genetic algo-rithm [Gerstoft et al., 2003b], a Markov–chain MonteCarlo sampling approach to estimate the uncertain-ties of the inverted parameters [Yardim et al., 2006],Markov state space model for microwave propaga-tion [Vasudevan et al., 2007], Kalman and particlefilters [Yardim et al., 2008], support vector machines[Douvenot et al., 2008], particle swarm optimization[Wang et al., 2009], a Bayesian approach with meteo-rological prior [Yardim et al., 2009], an improved best

fit approach [Douvenot and Fabbro, 2010; Douvenotet al., 2010] and a range adaptive objective function[Zhang et al., 2011a].

Gingras et al. [1997] suggested a matched-fieldprocessing approach for source localization and inver-sion for environmental parameters which was basedon plotting ambiguity surfaces of unknown variables.Gerstoft et al. [2000] showed successful application ofthe matched-field processing technique to invert forsurface-based duct parameters. They also showedthat it was not possible to invert for elevated ductparameters using single surface measurements.

Most of the previous RFC studies inverted theclutter power for the refractivity structure in a shortrange interval assuming changes in the refractivityprofile to be negligible. Gerstoft et al. [2003b] in-verted for a range-dependent profile by consideringrange-dependent parameters. Vasudevan et al. [2007]used a Markov chain model on the propagation statespace [Rabiner , 1989] to consider a range dependentprofile. The latter approach reduces the complex-ity of inversions based on the number of unknownprofiles with the added advantage of correcting in-verted profile of shorter ranges efficiently by consid-ering clutter power from longer ranges.

4.1. Likelihood function

The relationship between the observed complex-valued radar I and Q components of the field uI,oand uQ,o over Nr range bins and the predicted fielduI,s and uQ,s is described by the model:

uI,o =√

n1uI,s(m)ejφ1 + n2ejφ2 (25)

uQ,o =√

n1uQ,s(m)ejφ1 + n2ejφ2 (26)

where n1 is the multiplicative random variable in themodeled electric field due to a variable sea surfacereflectivity. Yardim et al. [2009] considered differ-ent probability distributions for the random variablen1 including lognormal, K-distribution and Rayleigh.Here, a lognormal distribution is assumed for each el-ement of the vector n1. Noise in the receiver, n2 andn2, are modeled by Gaussian distributions. φ1, φ2, φ2

are the random phase components of the complexrandom variable with uniform distributions:

{log n1}N1 , ∼ G(0, σ21) (27)

{n2}N1 , {n2}N1 ∼ G(0, σ22) (28)

{φ1}N1 , {φ2}N1 , {φ2}N1 ∼ U(0, π) (29)

The radar output power is obtained by:

Π = |uI |2 + |uQ|2. (30)


Thus, the observed and simulated clutter power arerelated by:

Πo = n1Πs(m) + nr (31)

{log n1}N1 ∼ G(0, σ21) (32)

{nr}N1 ∼ χ2 (33)

where n1 is the multiplicative noise with a lognor-mal distribution, and nr is the additive receiver noisewith a χ2 distribution and 2 degrees of freedom.Working in the high CNR (clutter to noise ratio)regime, the nr term can be neglected. Thus, themodeled power in the logarithmic domain is obtainedas:

Po = Ps(m) + n (34)

{n}N1 ∼ G(0, σ2), (35)

where, Po and Ps(m) are vectors of the observedand simulated clutter power of the profile m in dB,and n = 10 log n1.

More than one source of clutter power observa-tions can be used in an inversion. These sources caninclude the clutter power at different frequencies, dif-ferent radar elevation angles, or different snapshotswith similar conditions where Pn,o corresponds tothe nth source of the observed clutter power. GivenN different sources with uncorrelated noise power νn,the maximum likelihood function becomes: 1

L(m) =

N∏n=1

(πνn)−Nr exp

[−‖Po,n −Ps,n(m)‖2

νn

].

(36)Assuming that the noise power {νn}n=1..N is con-

stant across different observations, the negative log-likelihood function is simplified to

Φ(m) = − logL(m) ∝N∑n=1

‖Po,n−Ps,n(m)‖2. (37)

The maximum likelihood estimate m for m is ob-tained by minimizing (37) over the model parametervector m, which is similar to (23).

4.2. An inversion example

A set of refractivity profile measurements andradar returns was recorded at Wallops Island, Vir-ginia, April 1998 [Rogers et al., 2000; Gerstoft et al.,2003b]. Clutter signals were measured using theSpace Range Radar (SPANDAR) with operationalfrequency of 2.84 GHz, horizontal beamwidth of 0.4◦,elevation angle of 0, antenna height of 30.78 m, and

vertical polarization. The refractivity profiles of theenvironment were recorded using an instrumentedhelicopter provided by the Johns Hopkins UniversityApplied Physics Laboratory. The helicopter flew inand out along the 150◦ radial from a point 4 km dueeast of the SPANDAR in a saw-tooth pattern witheach transect lasting 30 min.

The range-dependent refractivity profile measuredby the helicopter is shown in Fig. 7a. This profile cor-responds to the measurement on April 2, 1998 from13:19:14 to 13:49:00 (Run 07). The spatial varia-tion of the M-profile is small in the 0–45 km range.Thus, RFC results of the corresponding clutter ob-servations are compared to the average of the mea-sured M-profiles in that range interval. Note thatalthough the experimental measurements are froma range-dependent refractivity profile, inversions arebased on a range-independent profile.

Recorded clutter power of the SPANDAR betweenazimuth 142–166◦ is used to estimate the trilinearfunction representing a surface-based duct since theclutter pattern (Fig. 1d) is rather stationary in thisinterval. The probability distribution of the refrac-tivity profile from all inversion results is obtained andthe maximum a posterior (MAP) solution of this dis-tribution is found to be the refractivity profile thatfits all data. Only the first 60 km of the radar clutteris used to invert for the refractivity profile to main-tain a high CNR and to avoid high spatial variationsof refractivity with range. A multiple angle clutter

Alt

itude

(m)

(a)

20 40 60 800

100

200

!40

!20

0

Range (km)

Alt

itude

(m)

(b)

20 40 60 800

100

200

!40

!20

0

(dB)!

Figure 6. Propagation loss: (a) MAP estimate of the re-fractivity profile given the clutter power at 150◦ azimuthof SPANDAR Run 07, and (b) standard atmosphere.

1 |x| = (|x1|, |x2|, ...) and ‖x‖2 =∑

i |xi|2.


0 10 20 30 40 50 600

50

100

150

200(a)340 380M

Alti

tude

(m)

300 3200

20

40

60

80

100

120

140

160

180

200

M−Profile

Alti

tude

(m)

(b)

Avg. heli. profile150° inversion

Range (km)

Clut

ter p

ower

(dB)

(c)

10 20 30 40 50 60−20

−10

0

10

20

30Clutter at 150°

Inverted profile of 150°

Figure 7. (a) Range-dependent refractivity profile recorded by an instrumented helicopter alongthe 150◦ azimuth. (b) Average of the first 45 km of the measured profile compared to the invertedprofiles of 150◦ clutter (solid) and the MAP profile of 142–166◦ (shaded). (c) observed and modeledclutter power of the inverted profile.

model based on curved wave beamforming [Karim-ian et al., 2011b] is used to calculate the clutterpower, and APM [Barrios, 2003] is used to calculatethe electric field and propagation loss. Fig. 7 showsthe inverted profiles obtained from clutter power ob-served along the 150◦ azimuth, the helicopter mea-sured refractivity along the 150◦ azimuth and thespan of inverted profiles using clutter power along142–166◦.

Fig. 6 shows the propagation loss using the in-verted profile from Fig. 7b and a standard atmo-sphere. Surface-based ducting conditions result in

the extended range of the radar and radar fadesin unexpected locations assuming a standard atmo-sphere. Radar parameters in this figure are identicalto those of the SPANDAR.

4.3. Bayesian approach

One important motivation behind estimation ofthe refractivity structure in the environment is topredict the radar performance in non-standard atmo-spheric conditions. This requires the statistical prop-erties of the parameters-of-interest such as the prop-agation loss which can be computed from the statis-


tical properties of the atmospheric refractivity. Theunknown environmental parameters are taken as ran-dom variables with corresponding one–dimensional(1–D) probability density functions (pdfs) and an n–dimensional joint pdf. This probability function canbe defined as the probability of the model vector mgiven the observed clutter power Po, p(m|Po), and itis called the posterior pdf (PPD). The profile m withthe highest probability is referred to as the maximuma posteriori (MAP) solution. The posterior means,variances, and marginal probability distributions canbe found by integrating over this PPD:

µi =

∫...

∫m′m′ip(m

′|Po)dm′, (38)

σ2i =

∫...

∫m′

(m′i − µi)2p(m′|Po)dm′, (39)

p(mi|Po) =

∫...

∫m′δ(m′i −mi)p(m

′|Po)dm′.(40)

The posterior density of any specific environmentalparameter can be obtained by marginalizing the n–dimensional PPD as given in (7) [Kay , 1993]. Ger-stoft et al. [2004] used importance sampling (IS)[MacKay , 2003] to compute the necessary multi-dimensional integrals needed to map the environmen-tal uncertainty into propagation loss uncertainty. ISproduces unbiased distributions of the desired vari-ables, however, the variance of the estimates dependheavily on the importance density used in IS. An-other problem with IS is the slow rate of convergencefor the numerical computation of the integrals. Ger-stoft et al. [2004] also compared IS to using just the1-D marginals of refractivity parameters to computethe PDF of propagation loss. As long as the interpa-rameter correlations are negligible, using marginalsis computationally more efficient than IS. They latershowed that lowering the peak clutter to noise ratiobroadens the a posteriori distribution of the propa-gation loss [Rogers et al., 2005].

The error in IS is minimized when samples aredrawn from the posterior distribution of the en-vironmental parameters p(m|Po). Sampling fromthe posterior requires a Markov chain Monte Carlo(MCMC) class sampler [O Ruanaidh and Fitzger-ald , 1996; MacKay , 2003] such as the Metropolis-Hastings (MH) [Metropolis et al., 1953] and theGibbs samplers [Geman and Geman, 1984]. MCMCmethods are guaranteed to asymptotically convergeto the true parameter distribution at a high compu-tational cost. Yardim et al. [2006] used a MH samplerto find the a posteriori distribution for the environ-

mental model parameters and used the MH sampleroutput to map the environmental uncertainty intothe propagation loss domain.

Yardim et al. [2007] introduced a hybrid genetic al-gorithms (GA)–MCMC method to estimate the pos-terior probability faster than MCMC which does notsuffer from the bias of histograms obtained from theGA. The hybrid GA–MCMC approximates the pos-terior distribution faster than an MCMC by firstperforming a GA inversion, discretizing the envi-ronmental parameter domain using the GA samplesvia Voronoi decomposition and the nearest neighbor-hood method [Sambridge, 1999a, b], and finally ap-plying a fast Gibbs sampler over this discrete space.The posterior distribution can be found using theBayes rule:

p(m|Po) =p(m)L(m)

p(Po)∝ p(m)L(m), (41)

with

p(Po) =

∫m

p(Po|m)p(m)dm. (42)

The likelihood function L(m) is the same as in(36), assuming a zero-mean Gaussian distribution forthe error. The prior p(m) represents a priori knowl-edge about the environmental parameters m, whichmight be from the meteorological statistics [Yardimet al., 2009] or from the result of previous inversions[Yardim et al., 2008; Douvenot et al., 2010]. A non-informative or flat prior assumption reduces (42) to:

p(m|Po) ∝ L(m), (43)

which has been discussed in Section 4.1. Fig. 8 isadopted from [Yardim et al., 2006] which shows thehighest posterior density (HPD) of the propagationloss obtained from the Metropolis samples of refrac-tivity model parameters from Fig. 7. Posterior dis-tributions are shown at a fixed range of 60 km anddifferent altitudes of 28 and 180 m, one inside andone outside the duct. The point inside the duct ex-hibits a narrow distribution while the variance of theestimated propagation loss outside the duct is muchlarger. As expected, the detection range increasesalong the horizon but this increase is not uniform.Fig. 8c shows the effects of uncertainty in the envi-ronmental parameters to a simple problem of targetdetection given that the target is an isotropic an-tenna with the radar cross section of 1 m2. The de-tection threshold in this example is chosen as 35 dBone way loss of the electric field.


−30 −20 −10 0 100

0.05

0.1

0.15

0.2

Propagation Factor, F (dB)

(a)

PPD(

F)

−30 −20 −10 0 100

0.05

0.1

0.15

0.2

PPD(

F)

Propagation Factor, F (dB)

(b)

10 20 30 40 50 60

20406080

100120140160180200

Range (km)

Heig

ht (m

)

(c)

90%80%70%

HPD Region

Figure 8. Posterior probability distribution of the propagation loss at range 60 km and altitudesof (a) 28 m, and (b) 180 m above the mean sea level, from the inversion of Fig. 7. (c) Detectionprobability given an isotropic target with an RCS of 1 m2.

A Markov state space model as discussed by Va-sudevan et al. [2007] also provides a Bayesian frame-work by considering the inversion result of the pre-vious states to invert for the current range step.

Continuous temporal and spatial variations in theenvironment led Yardim et al. [2008] to use extended[Kay , 1993] and unscented [Julier et al., 2000; Wanand van der Merve, 2001] Kalman filters to trackRFC results along with Sequential Monte Carlo [Gor-don et al., 1993; Yardim et al., 2011] methods suchas the particle filters. The paper compared the filterperformances in RFC tracking for different types ofducts and computed the Bayesian Cramer-Rao lowerbound (CRLB) which presents a lower bound to theRMS error.

Douvenot et al. [2010] provided a non-Bayesian ap-proach to inversion but modeled a history of inverted

parameters of surface-based ducts to keep the resultssmooth in azimuthal variations. They considered alibrary of pre-computed propagation losses of candi-date profiles to find the one with the minimum dis-tance to the observed clutter. Duct height variationsare limited in the latter study and a smoothing pro-cedure on the refractivity profiles is performed afterinversions.

4.4. Alternative RFC formulations

The form of the objective function in (37) suggeststhat some observations can be weighted more heavily.Usage of different frequencies is discussed in [Gerstoftet al., 2000]. Gerstoft et al. [2003a] argued that us-ing a single elevation angle results in inversions withlow precision above the duct height. Thus, they usedmultiple elevation angles of the radar with different


weights in the objective function to obtain more ro-bust inversions.

Rogers et al. [2005] considered a weighting for theclutter power according to the distance of the rangebin from the radar in an evaporation duct. Zhanget al. [2011a] suggested using an adaptive weightingalgorithm for different range bins in an evaporationduct that depends on the CNR. Rogers et al. [2005]have also suggested that RFC should be insensitive tothe small variations of peak locations of clutter powerwith range. Thus, they produced random replica ofthe predicted field Ps to make predictions less proneto the measurement errors.

Consideration of an l2 norm for error of Φ(m) =‖Po −Ps(m)‖2 is a consequence of assuming an ad-ditive uncorrelated Gaussian noise in (34). The termnr in (31) models the noise floor in the receiver whichhas been modeled by a linear truncation procedure inthe logarithmic power domain by Rogers et al. [2005]and by a complex Gaussian distribution on the fieldby Vasudevan et al. [2007]; Karimian et al. [2011a].A discussion of different random distributions andtheir effect on RFC is provided in [Yardim et al.,2009].

Other objective functions have also been suggestedin the statistical learning community. l1 (sum ofabsolute error terms) and the Huber norm [Huber ,1973] are less sensitive to the outliers than the com-monly used l2 norm. The Huber norm is a hybrid ofsmooth l2 norm for small errors and robust l1 treat-ment of large residuals, which has been used by Gui-tton and Symes [2003]; Ha et al. [2009] for the robustinversion of the seismic data.

There have been approaches that do not use theclutter equation as a forward model for inversions.Barrios [2004] used a rank correlation approach onthe ray tracing results of candidate profiles to invertfor the surface-based duct parameters based on theobserved clutter power of a 5.6 GHz radar. A to-mographic approach using a receiver array at the X-band and correlating the arrival wavefront spectrumto ray traces of candidate profiles has been suggestedby Zhao and Huang [2011]. In a similar problem,Park and Fabry [2011] used radar ground echo atlow elevation angles to estimate the vertical gradientof refractivity near the ground. They used ray trac-ing to model the radar coverage. One shortcoming ofthe current RFC approaches is evident when surfaceand weather (volume) clutter are hard to separatesuch as in precipitation.

5. Conclusion and future directions

RFC is an approach to estimate the refractivitystructure of a maritime environment based on theobserved radar clutter power. Marine ducts andtheir mathematical models have been discussed, anda framework for casting an inverse problem was pre-sented. An inversion consists of a forward modelto map the candidate profiles to the observation do-main, and a similarity measure to find the best pro-file. However, there are several shortcomings in thecurrent approaches to RFC that need to be addressedin future studies:

Bilinear and trilinear approximations to surface-based ducts are not representative of the duct struc-ture in some situations, and their performance wors-ens their performance worsens as the operational fre-quency increases. There have been attempts to over-come this problem by suggesting environmental re-fractivity models that rely on finding basis vectorsof the refractivity profile. Models for duct structuresare required that are simple (for easy inversion), andat the same time more representative of the true wavepropagation, especially if RFC is to be implementedat frequencies higher than 3 GHz.

Sea surface reflectivity models that are currentlyused in the radar community, e.g. the GIT model, donot represent well the sea reflections at very low graz-ing angles. Thus, remote sensing problems requiremore realistic models of the sea surface reflectivityat these angles (< 1◦).

One of the caveats of RFC algorithms is that de-tection of elevated ducts is not possible since thetrapped electromagnetic waves do not interact withthe sea surface. However, these ducts can be pre-dicted based on meteorological conditions [Gossard ,1981]. The 3-D refractivity profiles are intimatelylinked to the weather. There have been attempts toinclude climatological statistics of duct heights basedon the observation location and time of the year forevaporation ducts [Yardim et al., 2009].

Fusion of weather prediction algorithms with RFCinversions can greatly increase the performance ofboth. An example is in costal regions when the warmflow of air over the sea forms a rising surface ductfor radar propagation. Numerical weather prediction(NWP) systems have undergone substantial develop-ment in the last decade. There currently exist capa-bilities to extract 48 h radar forecast based on outputfrom NWP [Marshall et al., 2008]. These forecastsare used now to predict the radar performance [Le-


Furjah et al., 2010]. An improvement of RFC thenwould be using these fields as prior into the RFCinversion. After the inversion, the RFC posterior re-fractivity estimates could be used to influence thesmall-scale data assimilation for NWP. More researchis required to fill the gap between weather predictionand RFC.

Acknowledgments. This work was supported bySPAWAR under grant number N66001-03-2-8938, TDL0049. Authors would like to thank Dominique Lesselierand Ted Rogers for their constructive comments.

References

Anderson, K. D., Tropospheric refractivity profiles in-ferred from low elevation angle measurements ofGlobal Positioning System (GPS) signals, in AGARDConf. Proc., pp. 2.1–2.7, Bremerhaven, Germany,1994.

Anderson, K. D., Radar detection of low-altitude targetsin a maritime environment, IEEE Trans. AntennasPropagat., 43 (6), 609–613, doi:10.1109/8.387177,1995.

Babin, S. M., G. A. Young, and J. A. Carton,A new model of the oceanic evaporation duct,J.Appl.Meteor., 36 (3), 193–204, doi:10.1175/1520-0450(1997)036<0193:ANMOTO>2.0.CO;2, 1997.

Barrios, A. E., Parabolic equation modeling in hor-izontally inhomogeneous environments, IEEETrans. Antennas Propagat., 40 (7), 791–797,doi:10.1109/8.155744, 1992.

Barrios, A. E., A terrain parabolic equationmodel for propagation in the troposphere,IEEE Trans. Antennas Propagat., 42 (1), 90–98,doi:10.1109/8.272306, 1994.

Barrios, A. E., Advanced propagation model (APM)computer software configuration item (CSCI), Spaceand Naval Warfare Systems Center, TD 3145, 2002.

Barrios, A. E., Considerations in the development of theadvanced propagation model (APM) for U.S. Navy ap-plications, in International Conf. on Radar, Adelaide,Australia, 2003.

Barrios, A. E., Estimation of surface-based duct param-eters from surface clutter using a ray trace approach,Radio Sci., 39, RS6013, doi:10.1029/2003RS002930,2004.

Barton, D. K., Modern Radar System Analysis, ArtechHouse, 1988.

Blake, L. V., Radar Range Performance Analysis, Lex-ington Books, Lexington, MA, 1980.

Boyer, D., G. Gentry, J. Stapleton, and J. Cook, Us-ing remote refractivity sensing to predict troposphericrefractivity from measurements of propagation, inProc.AGARD SPP Symp. Remote Sensing: Valu-able Source Inform., pp. 16.1–16.13, Toulouse, France,1996.

Brooks, I. M., A. K. Goroch, and D. P.Rogers, Observations of strong sur-face radar ducts over the Persian Gulf,J.Appl.Meteor., 38 (9), 1293–1310, doi:10.1175/1520-0450(1999)038<1293:OOSSRD>2.0.CO;2, 1999.

Budden, K. G., The Wave-Guide Mode Theory of WavePropagation, Prentice-Hall, Englewood Cliffs, NJ,1961.

Dockery, G. D., Method for modeling sea surface clut-ter in complicated propagation environments, in IEEProc. Radar and Signal Processing, vol. 137, pp. 73–79,1990.

Dockery, G. D., and J. R. Kuttler, An improvedimpedance-boundary algorithm for Fourier split-step solution of the parabolic wave equation,IEEE Trans. Antennas Propagat., 44 (12), 1592–1599,doi:10.1109/8.546245, 1996.

Dockery, G. D., R. S. Awadallah, D. E. Freund, J. Z.Gehman, and M. H. Newkirk, An overview of recentadvances for the TEMPER radar propagation model,in IEEE Radar Conference, pp. 896–905, 2007.

Dosso, S. E., and J. Dettmer, Bayesian matched-field geoacoustic inversion, Inverse Problems, 27(5),055009, doi:10.1088/0266-5611/27/5/055009, 2011.

Dougherty, H., and B. A. Hart, Recent progress in ductpropagation predictions, IEEE Trans. Antennas Prop-agat., 27 (4), 542–548, doi:10.1109/TAP.1979.1142130,1979.

Douvenot, R., and V. Fabbro, On the knowledge ofradar coverage at sea using real time refractivityfrom clutter, IET Radar Sonar Navig., 4 (2), 293–301,doi:10.1049/iet–rsn.2009.0073, 2010.

Douvenot, R., V. Fabbro, P. Gerstoft, C. Bourlier, andJ. Saillard, A duct mapping method using least squaresupport vector machines, Radio Sci., 53, RS6005,doi:10.1029/2008RS003842, 2008.

Douvenot, R., V. Fabbro, P. Gerstoft, C. Bourlier, andJ. Saillard, Real time refractivity from clutter using abest fit approach improved with physical information,Radio Sci., 45, RS1007, doi:10.1029/2009RS004137,2010.

Doviak, R. J., and D. S. Zrnic, Doppler radar and weatherobservations, p. 562, 2nd ed., Academy Press, 1993.

Fabry, F., C. Frush, I. Zawadzki, and A. Kilambi, Onthe extraction of near-surface index of refraction us-ing radar phase measurements from ground targets,J.Atmos.Oceanic Technol., 14, 978–987, 1997.

Fairall, C. W., E. F. Bradley, D. P. Rogers, J. B. Ed-son, and G. S. Young, Bulk parametrization of air-seafluxes for tropical ocean-global atmosphere coupled-ocean atmosphere response experiment, J. Geophys.Res., 101 (C2), 3747–3764, doi:10.1029/95JC03205,1996.

Feit, M. D., and J. A. Fleck, Light propagationin graded-index fibers, Appl. Opt., 17, 3990–3998,doi:10.1364/AO.17.003990, 1978.

Fletcher, C., Clutter subroutine, Technology ServiceCorp., Silver Spring, MD, USA, Memorandum TSC-W84-01/cad, 1978.

Frederickson, P. A., K. L. Davidson, and A. K. Goroch,Operational bulk evaporation duct model for MO-RIAH, ver. 1.2, Tech. Rep. NPS/MR-2000-002, Nav.Postgrad. Sch., Monterey, Calif., 2000a.

Frederickson, P. A., K. L. Davidson, C. R. Zeisse,and C. S. Bendall, Estimating the refractive in-dex structure parameter (C2

n) over the ocean usingbulk methods., J. Appl. Meteor., 39, 1770–1783, doi:10.1175/1520-0450-39.10.1770, 2000b.


Geman, S., and D. Geman, Stochastic relaxation, Gibbsdistributions, and the Bayesian restoration of images,IEEE Trans. Pattern Anal.Machine Intell., 6, 721–741, doi:10.1080/02664769300000058, 1984.

Gerstoft, P., D. F. Gingras, L. T. Rogers, andW. S. Hodgkiss, Estimation of radio refractiv-ity structure using matched–field array processing,IEEE Trans. Antennas Propagat., 48 (3), 345–356,doi:10.1109/8.841895, 2000.

Gerstoft, P., L. T. Rogers, W. S. Hodgkiss, and L. J.Wagner, Refractivity estimation using multiple el-evation angles, IEEE J.Ocean. Eng., 28, 513–525,doi:10.1109/JOE.2003.816680, 2003a.

Gerstoft, P., L. T. Rogers, J. L. Krolik, andW. S. Hodgkiss, Inversion for refractivity parame-ters from radar sea clutter, Radio Sci., 38 (3), 1–22,doi:10.1029/2002RS002640, 2003b.

Gerstoft, P., W. S. Hodgkiss, L. T. Rogers, andM. Jablecki, Probability distribution of low-altitudepropagation loss from radar sea clutter data, RadioSci., 39, RS6006, doi:10.1029/2004/RS003077, 2004.

Gingras, D. F., P. Gerstoft, and N. L. Gerr, Electromag-netic matched field processing: basic concepts and tro-pospheric simulations, IEEE Trans. Antennas Propa-gat., 42 (10), 1536–1545, doi: 10.1109/8.633863, 1997.

Gordon, N. J., D. J. Salmond, and A. F. M. Smith, Novelapproach to nonlinear/non-Gaussian Bayesian stateestimation, IEE Proc. F, Radar and Signal Processing,140 (2), 107–113, doi:10.1049/ip-f-2.1993.0015, 1993.

Gossard, E. E., Clear weather meteorological effects onpropagation at frequencies above 1 GHz, Radio Sci.,16 (5), 589–608, doi:10.1029/RS016i005p00589, 1981.

Greenaert, G. L., Notes and correspondence on theevaporation duct for inhomogeneous conditions incoastal regions, J.Appl.Meteor. Climetol., 46 (4), 538–543, 2007.

Gregers-Hansen, V., and R. Mital, An empirical sea clut-ter model for low grazing angles, in IEEE Radar Con-ference, pp. 1–5, Pasadena, CA, 2009.

Guinard, N., J. Ransone, D. Randall, C. Purves,and P. Watkins, Propagation through an elevatedduct: Tradewinds III, IEEE Trans. Antennas Propa-gat., 12 (4), 479–490, doi:10.1109/TAP.1964.1138252,1964.

Guitton, A., and W. W. Symes, Robust inversion of seis-mic data using the Huber norm, Geophysics, 68 (4),1310–1319, doi:10.1190/1.1598124, 2003.

Ha, T., W. Chung, and C. Shin, Waveform inversion us-ing a back propagation algorithm and a Huber func-tion norm, Geophysics, 74 (3), R15–R24, 2009.

Haack, T., and S. D. Burk, Summertime ma-rine refractivity conditions along coastal Califor-nia, J.Appl.Meteor., 40, 673–687, doi:10.1175/1520-0450(2001)040<0673:SMRCAC>2.0.CO;2, 2001.

Hardin, R. H., and F. D. Tappert, Application of thesplit-step Fourier method to the numerical solutionof nonlinear and variable coefficient wave equations,SIAM Rev. 15, 423, 1973.

Helvey, R. A., Radiosonde errors and spurious surface-based ducts, Proc. IEE F, 130 (7), 643–648, 1983.

Hitney, H. V., Remote sensing of refractivity structure bydirect measurements at UHF, in AGARD Conf. Proc.,pp. 1.1–1.5, Cesme, Turkey, 1992.

Hitney, H. V., Refractive effects from VHF to EHF. PartA: Propagation mechanisms, in AGARD Lecture Se-ries, vol. LS-196, pp. 4A1–4A13, 1994.

Hitney, H. V., and J. H. Richter, Integrated Refrac-tive Effects Prediction System (IREPS), Nav. Eng. J.,88 (2), 257–262, 1976.

Horst, M., F. Dyer, and M. Tuley, Radar sea cluttermodel, in Int. IEEE AP/S URSI Symposium, vol.Proc. Part 2 (A80-15812 04-32), pp. 6–10, London In-stitute of electrical engineers, London, England, 1978.

Hua, Y., and W. Liu, Generalized Karhunen-Loeve trans-form, IEEE Signal Process. Lett., 5 (6), 141–142, 1998.

Huber, P. J., Robust regression: Asymptotics, conjec-tures, and Monte Carlo, Ann. Statist., 1 (5), 799–821,1973.

ITU reports, Report 1008-1, Reflection from the surfaceof the Earth, in Recommendations and reports of theCCIR, Propagation in nonionized media, vol. 5, 1990.

Janaswamy, R., Radio wave propagation over a non-constant immittance plane, Radio Sci., 36, 387–405,doi:10.1029/2000RS002338, 2001.

Jensen, F. B., W. A. Kuperman, M. B. Porter, andH. Schmidt, Computational ocean acoustics, SpringerLondon, Limited, 2011.

Jeske, H., State and limits of prediction methods forradar wave propagation conditions over the sea, inModern topics in Microwave Propagation and Air-SeaInteraction, edited by A. Zancla, pp. 130–148, D. Rei-del Publishing, 1973.

Julier, S., J. Uhlmann, and H. F. Durrant-White,A new method for nonlinear transformation ofmeans and covariances in filters and estima-tors, IEEE Trans. Automat. Control, 45, 477–482,doi:10.1109/9.847726, 2000.

Karimian, A., C. Yardim, P. Gerstoft, W. S. Hodgkiss,and A. E. Barrios, Estimation of refractivity using amultiple angle clutter model, Radio Sci., submitted,2011a.

Karimian, A., C. Yardim, P. Gerstoft, W. S. Hodgkiss,and A. E. Barrios, Multiple grazing angle sea cluttermodeling, IEEE Trans. Antennas Propagat., submit-ted, 2011b.

Katzin, M., R. W. Bauchman, and W. Binnian, 3- and 9-centimeter propagation in low ocean ducts, Proc. IRE,35, 891–905, 1947.

Kay, S. M., Fundamentals of statistical signal processing,volume I: Estimation theory, Prentice Hall, EnglewoodCliffs, NJ, 1993.

Kerr, D. E., Propagation of short radio waves, McGraw-Hill, 1951.

Ko, H. W., J. W. Sari, and J. P. Skura, Anomalousmicrowave propagation through atmospheric ducts,Johns Hopkins APL Tech. Dig. 4(2), 12–26, 1983.

Konstanzer, G. C., J. Z. Gehman, M. H. Newkirk, andG. D. Dockery, Calculation of the surface incident fieldusing TEMPER for land and sea clutter modeling,Proc. on Low Grazing Angle Clutter: Its Character-ization, Measurement and Application, RTO-MP-60,pp. 14–1–14–2, 2000.


Kraut, S., R. Anderson, and J. Krolik, A gener-alized Karhunen-Loeve basis for efficient esti-mation of tropospheric refractivity using radarclutter, IEEE Trans. Sig. Proc., 52 (1), 48–59,doi:10.1109/TSP.2003.820297, 2004.

Krolik, J. L., and J. Tabrikian, Tropospheric refractivityestimation using radar clutter from the sea surface, inProc. Battlespace Atmospheric Conference, SSC-SD,pp. 635–642, San Diego, CA, 1997.

Kukushkin, A., Radio wave propagation in the marineboundary layer, Wiley, 2004.

Kuttler, J. R., Differences between the narrow-angleand wide-angle propagators in the split-step Fouriersolution of the parabolic wave equation, IEEETrans. Antennas Propagat., 47 (7), 1131–1140, doi:10.1109/8.785743, 1999.

Kuttler, J. R., and R. Janaswamy, Improved Fouriertransform methods for solving the parabolic waveequation, Radio Sci., 37 (2), 1021–1031, doi:10.1029/2001RS002488, 2002.

LeFurjah, G., R. Marshall, T. Casey, T. Haack, andD. De Forest Boyer, Synthesis of mesoscale numeri-cal weather prediction and empirical site-specific radarclutter models, IET Radar Sonar Navig., 4 (6), 747–754, 2010.

Leontovich, M. A., and V. A. Fock, Solution of the prob-lem of electromagnetic wave propagation along theearth’s surface by the method of parabolic equation,Acad. Sci. USSR J. Phys., 10, 13–23, 1946.

Levy, M., Parabolic equation methods for electromagneticwave propagation, The Institution of Electrical Engi-neers, London, 2000.

Lin, L., Z. Zhao, Y. Zhang, and Q. Zhu, Tropo-spheric refractivity profiling based on refractivity pro-file model using single ground-based global position-ing system, Radar Sonar Navigation, IET, 5 (1), 7–11,doi:10.1049/iet-rsn.2009.0167, 2011.

Liu, W. T., K. B. Kataros, and J. A. Businger, Bulk pa-rameterization of air-sea exchanges of heat and watervapor including the molecular constraints at the inter-face, J. Atmos. Sci., 36, 1722–1735, doi:10.1175/1520-0469(1979)036<1722:BPOASE>2.0.CO;2, 1979.

Lowry, A. R., C. Rocken, S. V. Sokolovsky, and K. D.Anderson, Vertical profiling of atmospheric refractiv-ity from ground-based GPS, Radio Sci., 37, 1041–1059,doi:10.1029/2000RS002565, 2002.

MacKay, D. J. C., Information Theory, Inference andLearning Algorithms, Cambridge University Press,Cambridge, United Kingdom, 2003.

Marshall, R. E., W. D. Thornton, G. LeFurjah, andS. Casey, Modeling and simulation of notional futureradar in non-standard propagation environments fa-cilitated by mesoscale numerical weather predictionmodeling, Naval Engineers Journal, 120 (4), 55–66,doi:10.1111/j.1559-3584.2008.00165.x, 2008.

Mentes, S., and Z. Kaymaz, Investigationof surface duct conditions over Istanbul,Turkey, J.Appl.Meteor. Climetol., 46, 318–337,doi:10.1175/JAM2452.1, 2007.

Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth,A. H. Teller, and E. Teller, Equation of state cal-culations by fast computing machines, J. of ChemicalPhysics, 21, 1087–1092, doi:10.1063/1.1699114, 1953.

Miller, A. R., R. M. Brown, and E. Vegh, New deriva-tion for the rough surface reflection coefficient and forthe distribution of the sea-wave elevations, Proc. Inst.Elect. Eng., 131 (2, pt.H), 114–116, doi:10.1049/ip-h-1.1984.0022, 1984.

Miller, E. L., and A. S. Willsky, A multiscale, statisticallybased inversion scheme for linearized inverse scatteringproblems, IEEE Trans.Geosci. Remote Sens., 34 (2),346–357, doi:10.1109/36.485112, 1996a.

Miller, E. L., and A. S. Willsky, Wavelet-based meth-ods for the nonlinear inverse scattering problem usingthe extended Born approximation, Radio Sci., 31 (1),51–65, doi:10.1029/95RS03130, 1996b.

Nathanson, F. E., J. P. Reilly, and M. N. Cohen, Radardesign principles: Signal processing and the environ-ment, 2 ed., McGraw-Hill, 1991.

O Ruanaidh, J. J. K., and W. J. Fitzgerald, NumericalBayesian Methods Applied to Signal Processing, Statis-tics and Computing Series, Springer–Verlag, NewYork, 1996.

Pappert, R. A., R. A. Paulus, and F. D. Tappert, Seaecho in tropospheric ducting environments, Radio Sci.,27 (2), 189–209, doi:10.1029/91RS02962, 1992.

Park, S., and F. Fabry, Estimation of near-groundpropagation conditions using radar ground echocoverage, J.Atmos.Oceanic Technol., 28, 165–180,doi:10.1175/2010JTECHA1500.1, 2011.

Patterson, W., Ducting climatology summary, Tech. rep.,SPAWAR Sys. Cent., San Diego, Calif., 1992.

Paulus, R. A., Practical applications of an evapo-ration duct model, Radio Sci., 20, 887–896, doi:10.1029/RS020i004p00887, 1985.

Paulus, R. A., Evaporation duct effects on sea clutter,Radio Sci., 38 (11), 1765–1771, doi:10.1109/8.102737,1990.

Pekeris, C. L., Accuracy of the earth flatten-ing approximation in the theory of microwavepropagation, Phys. Rev., 70 (7-8), 518–522,doi:10.1103/PhysRev.70.518, 1946.

Phillips, O. M., Spectral and statistical proper-ties of the equilibrium range in wind-generatedgravity waves, J. Flui. Mech., 156, 505–531,doi:10.1017/S0022112085002221, 1985.

Rabiner, L. R., A tutorial on hidden Markov mod-els and selected applications in speech recognition,Proc. IEEE, 77 (2), 257–286, doi:10.1109/5.18626,1989.

Reilly, J. P., and G. D. Dockery, Influence of evaporationducts on radar sea return, in IEE Proc. Radar and Sig-nal Processing, vol. 137, pp. 80–88, 1990.

Richter, J. H., Structure, variability, and sensing of thecoastal environments, in AGARD SPP Symposium onPropagation Assessment in Coastal Environments, pp.1.1–7.14, NATO AGARD, Neuilly-sur-Seine, France,1995.

Roberts, R. D., E. Nelson, J. W. Wilson, N. Re-hak, J. Sun, S. Ellis, T. Weckwerth, F. Fabry,P. Kennedy, J. Fritz, V. Chandrasekar, S. Reising,S. Padamanabhan, J. Braun, T. Crum, L. Mooney,and R. Palmer, REFRACTT 2006: Real time re-trieval of high-resolution, low-level moisture fieldsfrom operational NEXRAD and research radars,Bull. Amer.Meteor. Soc., 89 (10), 1535–1538, 2008.


Rogers, L. T., Effects of the variability of at-mospheric refractivity on propagation estimates,IEEE Trans. Antennas Propagat., 44 (4), 460–465,doi:10.1109/8.489297, 1996.

Rogers, L. T., Likelihood estimation of troposphericduct parameters from horizontal propagation measure-ments, Radio Sci., 32, 79–92, doi:10.1029/96RS02904,1997.

Rogers, L. T., and R. A. Paulus, Measured performanceof evaporation duct models, Tech. Rep. 2989, NavalCommand, Command and Ocean Surviellance CenterRDTE Div., 1996.

Rogers, L. T., C. P. Hattan, and J. K. Sta-pleton, Estimating evaporation duct heights fromradar sea echo, Radio Sci., 35 (4), 955–966,doi:10.1029/1999RS002275, 2000.

Rogers, L. T., M. Jablecki, and P. Gerstoft, Poste-rior distribution of a statistic propagation loss in-ferred from radar sea clutter, Radio Sci., 40, RS6005,doi:10.1029/2004RS003112, 2005.

Rowland, J. R., G. C. Konstanzer, M. R. Neves, R. E.Miller, J. H. Meyer, and J. R. Rottier, Seawasp:refractivity characterization using shipboard sensors,in Battlespace Atmospheric Conferene, pp. 155–164,RDT&E Div., Nav. Command Control and OceanSurv. Cent., San Diego, Calif., 1996.

Sambridge, M., Geophysical inversion with a neighbor-hood algorithm - I. searching a parameter space, Geo-phys. J. Int., 138, 479–494, 1999a.

Sambridge, M., Geophysical inversion with a neighbor-hood algorithm - II. appraising the ensemble, Geo-phys. J. Int., 138, 727–746, 1999b.

Schmidt, R. O., Multiple emitter location and signal pa-rameter estimation, IEEE Trans. Antennas Propagat.,34 (3), 276–280, doi:10.1109/TAP.1986.1143830, 1986.

Sittrop, H., Characteristics of clutter and target at X-and Ku- band, in AGARD Conf. Proc., no. 197 inNew Techniques and Systems in Radar, pp. 28.1–28.27,Hague, Netherlands, 1977.

Skolnik, M. I., Radar handbook, 3 ed., McGraw-Hill, NewYork, 2008.

Skura, J. P., C. E. Schemm, H. W. Ko, and L. P.Manzi, Radar coverage predictions through time- andrange-dependent refractive atmosphere with planetaryboundary layer and electromagnetic parabolic equa-tion models, in IEEE Radar Conference, pp. 370–378,1990.

Tabrikian, J., and J. Krolik, Theoretical perfor-mance limits on tropospheric refractivity estima-tion using point–to–point microwave measurements,IEEE Trans. Antennas Propagat., 47 (11), 1727–1734,doi:10.1109/8.814953, 1999.

Thomson, D. J., and N. R. Chapman, A wide-angle split-step algorithm for the parabolic equation, J. Acoust.Soc. Am., 74 (6), 1848–1854, doi:10.1121/1.390272,1983.

Vasudevan, S., R. Anderson, S. Kraut, P. Ger-stoft, L. T. Rogers, and J. L. Krolik, Recur-sive Bayesian electromagnetic refractivity estimationfrom radar sea clutter, Radio Sci., 42, RS2014,doi:10.1029/2005RS003423, 2007.

Von Engeln, A. G., G. Nedoluha, and J. Teixeira, Ananalysis of the frequency and distribution of ductingevents in simulated radio occultation measurementsbased on ECMWF fields, J.Geophys. Res., 108 (D21,4669), doi:10.1029/2002JD003170, 2003.

Wakimoto, R. M., and M. H. Murphy, Analysis ofa dryline during IHOP: implications for convec-tion initiation, Mon.Wea.Rev., 137 (3), 912–936,doi:10.1175/2008MWR2584.1, 2009.

Wan, E. A., and R. van der Merve, “The unscentedKalman filter”, in S. Haykin, Kalman Filtering andNeural Networks, John Wiley & Sons, New York, 2001.

Wandinger, U., Raman Lidar, in Lidar, Springer Seriesin Optical Sciences, vol. 102, edited by C. Weitkamp,pp. 241–271, Springer Berlin / Heidelberg, 2005.

Wang, B., Z. S. Wu, A. Zhao, and H. G.Wang, Retrieving evaporation duct heights fromradar sea clutter using particle swarm optimiza-tion, Progress In Electromagnetics Research, 9, 79–91,doi:10.2528/PIERM09090403, 2009.

Weckwerth, T. M., C. R. Petter, F. Fabry, S. J.Park, M. A. LeMone, and J. W. Wilson, Radarrefractivity retrieval: Validation and application toshort–term forcasting, J.Appl.Meteor., 44, 285–300,doi:10.1175/JAM-2204.1, 2005.

Willitsford, A., and C. R. Philbrick, Lidar description ofthe evaporation duct in ocean environments, in SPIE,vol. 5885, pp. 140–147, Bellingham, WA, 2005.

Wilson, J. W., and R. D. Roberts, Summary of convec-tive storm initiation and evolution during IHOP: Ob-servational and modeling perspective, Mon.Wea.Rev.,134 (1), 23–47, doi:10.1175/MWR3069.1, 2006.

Yardim, C., P. Gerstoft, and W. S. Hodgkiss, Es-timation of radio refractivity from radar clut-ter using Bayesian Monte Carlo analysis, IEEETrans. Antennas Propagat., 54 (4), 1318–1327, doi:10.1109/TAP.2006.872673, 2006.

Yardim, C., P. Gerstoft, and W. S. Hodgkiss, Statisticalmaritime radar duct estimation using a hybrid geneticalgorithm—Markov chain Monte Carlo method, RadioSci., 42, RS3014, doi:10.1029/2006RS003561, 2007.

Yardim, C., P. Gerstoft, and W. S. Hodgkiss, Trackingrefractivity from clutter using Kalman and particle fil-ters, IEEE Trans. Antennas Propagat., 56 (4), 1058–1070, doi:10.1109/TAP.2008.919205, 2008.

Yardim, C., P. Gerstoft, and W. S. Hodgkiss, Sensitiv-ity analysis and performance estimation of refractiv-ity from clutter technique, Radio Sci., 44, RS1008,doi:10.1029/2008RS003897, 2009.

Yardim, C., Z. H. Michalopoulou, and P. Gerstoft,An overview of sequential Bayesian filtering inocean acoustics, IEEE J.Ocean. Eng., 36 (1), 71–89,doi:10.1109/JOE.2010.2098810, 2011.

Zhang, J. P., Z. S. Wu, and B. Wang, An adaptiveobjective function for evaporation duct estimationfrom radar sea echo, Chin. Phys. Lett., 28 (3, 034301),doi:10.1088/0256-307X/28/3/034301, 2011a.

Zhang, J. P., Z. S. Wu, Q. L. Zhu, and B. Wang, A four-parameter M-profile model for the evaporation ductestimation from radar clutter, Progress In Electromag-netics Research, 114, 353–368, 2011b.

Zhao, X. F., and S. X. Huang, Refractivity estimationsfrom an angle of arrival spectrum, Chin. Phys. B, 20 (2,029201), doi:10.1088/1674-1056/20/2/029201, 2011.


Zhao, X. F., S. X. Huang, and H. D. Du, Theoreticalanalysis and numerical experiments of variational ad-joint approach for refractivity estimation, Radio Sci.,46, RS1006, doi:10.1029/2010RS004417, 2011.

A. Karimian, C. Yardim, P. Gerstoft, WilliamS. Hodgkiss, Marine Physical Laboratory, ScrippsInst. of Oceanography, University of California,

San Diego, La Jolla, CA 92037–0238 USA ([email protected], [email protected], [email protected],[email protected]).

A. E. Barrios, Atmospheric Propagation Branch,Space and Naval Warfare Systems Center, San Diego,California, USA ([email protected])

(Received .)

Documents

RFC Review