remote sensing
Article Estimation of Surface Duct Using Ground-Based GPS Phase Delay and Propagation Loss
Qixiang Liao ID , Zheng Sheng *, Hanqing Shi, Jie Xiang and Hong Yu College of Meteorology and Oceanology, National University of Defense Technology, Nanjing 211101, China; [email protected] (Q.L.); [email protected] (H.S.); [email protected] (J.X.); [email protected] (H.Y.) * Correspondence: [email protected]; Tel.: +86-025-830-624
Received: 24 March 2018; Accepted: 7 May 2018; Published: 8 May 2018
Abstract: The propagation of Global Positioning System (GPS) signals at low-elevation angles is significantly affected by a surface duct. In this paper, we present an improved algorithm known as NSSAGA, in which simulated annealing (SA) is combined with the non-dominated sorting genetic algorithm II (NSGA-II). Matched-field processing was used to remotely sense the refractivity structure by using the data of ground-based GPS phase delay and propagation loss from multiple antenna heights. The performance was checked by simulation data with and without noise. In comparison with NSGA-II, the new hybrid algorithm retrieved the refractivity structure more efficiently under various noise conditions. We then modified the objective function and found that matched-field processing is more effective than the conventional least-squares method for inferring the refractivity parameters. Comparing the inversion results and in situ sounding data suggests that the improved method presented herein can capture refractivity characteristics in realistic environments.
Keywords: surface duct; refractivity structure; GPS; remote sensing; NSGA-II; matched-field processing
1. Introduction An atmospheric duct is an atmospheric layer in which strong vertical gradients of refractivity determined by temperature and humidity affect the propagation of electromagnetic waves. The electromagnetic waves are trapped within the ducting layer with weak propagation loss and enhanced propagation range, usually known as atmospheric ducting [1,2]. Turton et al. [3] presented a classification of different radio-propagation conditions and corresponding atmospheric ducting effects. They pointed out that ducting may be associated with the following five meteorological processes: (1) sea-surface evaporation, (2) anticyclonic subsidence, (3) subsidence of cold air beneath frontal surfaces, (4) nocturnal radiative cooling over land, and (5) advection. A strong temperature inversion is often present at the top of a well-mixed marine boundary layer in these meteorological processes [4]. Obviously, there exists a correlation between ducting and boundary-layer inversion. When warm and dry air masses from land travel over cooler water, there exists sharp differences of temperature and humidity vertically, which may result in ducting formation. Propagation loss for radar radio waves within a duct will usually be much less than that in a non-ducting atmospheric environment, and hence radar can detect targets at greater distances. A radar “hole” may also appear just above the duct because of short detection ranges [5]. These characteristics have significant impacts on radar target detection and target tracking because failure to consider a ducting environment may yield false radar returns. Therefore, the study of atmospheric ducts can not only improve the performance of radar or communications systems but can also contribute to research into the atmospheric boundary layer [6–9]. It is a difficult problem to quantify the patterns of an atmospheric duct in time and space. The methods of sounding refractivity profiles—such as radiosondes, microwave refractometers, and evaporation duct
Remote Sens. 2018, 10, 724; doi:10.3390/rs10050724 www.mdpi.com/journal/remotesensing Remote Sens. 2018, 10, 724 2 of 20 sensors—have been developed [10–14]. However, these methods are direct sensing techniques and have lower sparse sampling rates. Krolik and Tabrikian [15] first used radar clutter data to retrieve atmospheric refractivity profiles. Afterward, the technique of refractivity from clutter (RFC) was developed substantially.A variety of intelligent algorithms have been used to estimate refractivity profiles, for example, the genetic algorithm (GA) [16,17], the Bayesian-Markov-chain Monte Carlo method [18], the support vector machine [19], and others [20–23]. However, these algorithms have some defects in practical applications [24,25]. A promising approach for sensing refractivity profiles remotely is the use of Global Positioning System (GPS) signals [10,26]. Hitney [27] demonstrated the capability of estimating the trapping layer’s base height from ultrahigh frequency (UHF) signal strengths. Anderson [28] inferred the refractivity structure in the lower atmosphere based on the phase delay from a ground-based GPS receiver as the satellite rose or set on the horizon. Lowery et al. [29] and Lin et al. [30] estimated the refractivity structure using a ray-tracing model to fit measured GPS phase delays with least-squares and evaluated the potential and limitations of this method [31]. Subsequently, other scholars [32,33] also inferred refractivity profiles from measured excess phase path. However, only one type of observational data is used in these methods, such as the GPS signal strength or phase delay. Therefore, Wu et al. [34] tried to combine the GPS phase delay and bending angle to retrieve refractivity profiles, but they neglected the fact that the observed phase delay and bending angles are not independent. For this reason, Liao et al. [35] used the non-dominated sorting genetic algorithm II (NSGA-II) to retrieve the atmospheric refractivity structure based on the ground-based GPS phase delay and propagation loss. Although the combination of ground-based GPS phase delay and propagation loss can infer refractivity parameters effectively, its performance is limited by the weak global search capability of NSGA-II. Therefore, in the present paper, we propose an improved algorithm based on NSGA-II in which the method of electromagnetic match-field processing is applied [36]. Furthermore, we also integrate the GPS phase delay and propagation loss from multiple antenna heights to infer the refractivity parameters. The present paper is a sequel to Liao et al. [35] and is aimed at quantifying the performance of the joint inversion for two types of observation using improved NSGA-II. Here, the objective function is match-field processing. The remainder of this paper is organized as follows. Section2 introduces the forward model, including models of GPS phase delay and radio signal propagation. Section3 proposes a new algorithm—the non-dominated sorting and simulated annealing genetic algorithm (NSSAGA)—which is an improved version of NSGA-II that includes simulated annealing (SA), and it describes the algorithm implementation step by step. Section3 also introduces a simplified mathematical model of a surface duct and a related objective function. Section4 presents the NSSAGA algorithm used for refractivity-profile estimation. Finally, conclusions from this work are summarized in Section5.
2. Forward Model
2.1. Phase-Delay Model In this section, some basic concepts regarding the ground-based GPS phase-delay model are introduced briefly. Because of variations in atmospheric temperature T, pressure P, and water vapor pressure e, the atmospheric refractivity N is approximated by a formula due to Smith and Weintraub [37]: N = 77.6 × p/T + 3.73 × 105 × (e/T2) (1) where the unit of T is K and the unit of P and e is hPa. The modified refractivity M, which considers the Earth’s curvature, is related to the radio refractivity N as follows [38]:
M = N + 0.157 × r (2) Remote Sens. 2018, 10, x FOR PEER REVIEW 3 of 20 Remote Sens. 2018, 10, 724 3 of 20
M N 0.157 r (2) Differentiating with respect to r, we get Differentiating with respect to r , we get dMdM/dr =/ drdN/dNdr +/ dr0.157 0.157 (3) (3) wherewherer is ther is altitude the altitude in meters in andmetersM isand the modifiedM is the refractivity. modified refractivity. The unit of dMThe/ drunitis M-units/km.of dM / dr is EquationsM-units/km. (2) and Equations (3) are applied (2) and for (3) all are radio applied frequencies for all radio with frequencies little error. Referringwith little toerror. Almond Referring and to ClarkeAlmond [39], anddN/ Clarkedr—which [39], isdN the/ dr gradient—which of is refractivity the gradientN ofwith refractivity respect toN altitude with respectr—is usedto altitude to classifyr —is the used profiles to classify and its the unit profiles is N-units/km. and its unit Normally, is N-units/km. the atmospheric Normally, refractivity the atmospheric has a negativerefractivity gradienthas a whennegative the electromagneticgradient when the waves electromagnetic bend toward thewaves Earth. bend However, toward if the the Earth. refractivity However, gradient if the is lessrefractivity than −160 gradient N-units/km, is less anthan atmospheric −160 N-units/km, duct occurs. an atmospheric The atmospheric duct structure,occurs. The especially atmospheric a surfacestructure, duct, mayespecially delay a the surface GPS signal duct, atmay low delay elevation. the GPS Hence, signal this at study low usedelevation. the GPS Hence, phase this delay study to retrieveused the the GPS surface phase duct. delay to retrieve the surface duct. FigureFigure1 shows 1 shows the geometrythe geometry of the of raythe ray path path in a in ground-based a ground-based GPS GPS occultation, occultation, where whereR and R Tand r r O areT the are GPS the receiver GPS receiver and transmitter, and transmitter, respectively, respectively, and 1 and and2 are r1 the and distances r2 are fromthe distances the geocenter from the to thegeocenter receiver O and to transmitter, the receiver respectively. and transmitter, respectively.
α φ2 T φ1 R β
r2 r1 θ
O
Figure 1. Geometry of ground-based Global Positioning System (GPS) occultation for computing Figure 1. Geometry of ground-based Global Positioning System (GPS) occultation for computing phase-delay. phase-delay.
AssumingAssuming that that the atmospherethe atmosphere is a spherically is a sphericall symmetricy symmetric and layered and layered refractive refractive medium, medium,the GPS the signalGPS path signal is path approximated is approximated as a function as a function of r only. of r The only. phase The pathphase is path determined is determined according according to Shengto Sheng et al. [et32 al.]: [32]: r Z 2 S = n(r2)dl (4) S n(r)dl r1 (4) √ r1 where dl = dr2 + r2dθ2 is a differential length along the ray path and n is the refractive index with 6 2 2 2 N =where (n − 1dl) ×10 dr. Using r Bouger’sd is a law—namely, differential lengtha = nr alongsin φ ,the where ray apathis a and constant n is impact the refractive parameter index associated with the ray path—and6 substituting dl = dr/ cos φ, Equation (4) can be represented as: with N (n 1)10 . Using Bouger’s law—namely, a nr sin , where a is a constant impact r x parameter associatedZ 2 with the ray path—andZ 2 substituting dl dr / cos , Equation (4) can be 2 p 2 2 2 p 2 2 represented as:S = r · n / r · n − a dr = x · [1 − (x/n)dn/dx]/ x − a dx (5) r1 x1 r2 x2 2 2 2 2 2 2 where x = rn(r) denotesS ther n refractive/ r n radius. a dr The ray x path1 ( inx a/ n vacuum)dn / dx can/ bex determined a dx as: (5) r1 x1 q 2 2 S = r + r − r1 · r · where x rn(r) denotes the refractive0 1radius.2 The2 ray2 pacosthθ in a vacuum can be determined as:(6) where the angle θ is the spherical angle between 2r1 and2 r2. Hence, the excess phase path is defined as: S0 r1 r2 2r1 r2 cos (6)
∆S = S − S0 (7) where the angle is the spherical angle between r1 and r2 . Hence, the excess phase path is defined as:
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2.2. Propagation-Loss Model To use the propagation loss of ground-based GPS, a forward-propagation model is established. The electromagnetic wave propagation equation (PE), derived from the wave equation by the parabolic approximation, is widely used to model refractive effects on GPS signal propagation [40]. Teti [41] explains the PE theory in detail. Considering the spherical shape of the Earth, the two-dimensional (2D) narrow-angle forward PE can be written as:
∂u(x, z)/∂x = i/(2 · k) · ∂2u(x, z)/∂z2 + i · k/2 · (M2(x, z) − 1) · u(x, z) (8) where x and z represent the horizontal axis (range) and the vertical axis (height), respectively, k is the wavenumber in free space, M(x, z) is the modified refractivity, and u(x, z) is the electromagnetic field component at range x and height z. Equation (8) can be used for both horizontal and vertical polarization. If the initial field is provided, the split-step Fourier transform (SSFT) is used to calculate low-elevation GPS signal propagation in a tropospheric duct. This can be substituted by a sine or cosine transform because the bottom boundary approximates to a perfectly conducting surface. The SSFT solution of propagated wave equation is expressed as [26]: n o u(x + ∆x, z) = exp(ik(M2 − 1)∆x/2)F−1 exp(−iπ2 p2∆x/(2k))F[u(x, z)] (9) where F and F−1 are the Fourier transform and inverse Fourier transform, respectively. Here, ∆x is the range step, p = 2k sin θ represents the transform variable for which θ is the angle from the horizontal, and u(x, z) is the initial field. Balvedi and Walter [42] give a more detailed process for solving the parabolic equation. To model the GPS signal propagation, the single-way propagation loss L(x, z) of ground-based GPS signals is used to describe the objective function [18]:
L(x, z) = 20 log( f ) + 20 log(x) + C (10) where f represents the propagation factor and C is a constant parameter. In a rectangular coordinate system, the propagation can be calculated as: √ f = x|u(x, z)| (11)
3. Improved Inversion Algorithm
3.1. Proposed NSSAGA Algorithm Combining the GPS phase delay and propagation loss is a multiobjective optimization process. Hence, we used NSGA-II—a multiobjective optimization algorithm—to search for the best fit [35,43]. Considering that NSGA-II is an improvement of NSGA, one of the first evolutionary algorithms (EAs) for finding multiple Pareto-optimal solutions, the former retains premature convergence, which is a common problem with conventional GAs. In other words, the solving procedure becomes trapped at a local optimum and the inversion parameters converge to a suboptimal solution [44]. The SA algorithm was first proposed by Metropolis et al. [45]. It is based on the physical principle of heating a material and then slowly lowering the temperature to decrease defects, further minimizing the system energy [46]. SA is a probabilistic technique for approximating the global optimum of a given function. Combining SA and NSGA-II could overcome premature convergence effectively and find the global optimum instead of a local one [47]. Therefore, we combined them into a hybrid algorithm, namely, NSSAGA, that we used for surface-duct retrieval. Remote Sens. 2018, 10, 724 5 of 20
3.2. Implementation Steps Based on the above ideas, the procedure of the proposed hybrid algorithm is shown in Figure2. The detailed design steps for parameter estimation by the NSSAGA algorithm are realized as follows. Step 1: Initialize the parameter values of the hybrid algorithm. The population size is 200 and the maximum number of generations is set to 10. In NSSAGA, the population size is the number of candidate solutions and the number of generations is the number of program iterations. The numbers of objective functions and variable parameters in the program are set to 2 and 4, respectively. A variable parameter is a physical variable to be inverted. For the real-coded NSSAGA, a multi-parent extension of the simulated binary crossover (SBX) operator is used, and the distribution indexes for the crossover and mutation operators are set to ηc = 20 and ηm = 20, respectively. The crossover and mutation operators βi are defined by
( 1/(η+1) (2ui) i f ui ≤ 0.5 βi = 1/(η+1) (12) [1/(2(1 − ui))] Otherwise
Here, ui determines the probability of crossing or mutating and is chosen randomly from the interval [0,1], where i is the generation number. We set the initial temperature as To = 100 and the end temperature as Tend = 0. The cooling factor ρ is 0.8, determining the rate of change of temperature required in SA. Step 2: Initially, a parent population Pt = [x1, x2, x3, ··· , x200] is generated randomly within the boundaries of the solution while calculating the corresponding function value. The SA time i is zero. Step 3: Begin the main loop. If the SA condition Ti > Tend is met, we set the generation variable as t = 0. Otherwise, the program ends. Step 4: The usual binary tournament selection, cross operators, and mutation operators are used to create an offspring Qt = [x1, x2, x3, ··· , x200] with the corresponding function value. Here, the usual binary tournament selection involves running several “tournaments” among a few individuals chosen at random from the population. Step 5: The Metropolis criterion is the core of the SA algorithm, which derives from the importance-sampling method proposed by Metropolis et al. [45]. Designing the system energy E as the function value J, let ∆ = Jnew − Jold and move the system to the new state if the random variable p, distributed uniformly over (0,1), satisfies
p ≤ exp(−∆/Ti) (13) where Ti is the current temperature. Step 6: A combined population Rt = Pt ∪ Qt is formed, where Rt = 400. In a typical multiobjective optimization problem, there exists a set of solutions that are superior to the other solutions in the search space when all objectives are considered but are inferior to other solutions in the space in one or more objectives. These solutions are called as non-dominated solutions [46]. The population Rt is sorted based on the non-domination levels, and 200 individuals in the population Rt are chosen to be the new parent population Pt+1. We update the generation variable t = t + 1 and archive the best individual in the new population to the vector Pt,best. If the iteration condition meets t < Maxgen, repeat Steps 4–6. Otherwise, we achieve the best individual from Pt,best to Si,best and repeat Steps 3–6, updating the number i = i + 1. Step 7: Once we have completed Steps 3–6 and the SA condition Ti < Tend is met, the process stops, and the system is considered to have frozen to the lowest temperature as defined by the annealing schedule. Output the best of Si,best. Remote Sens. 2018, 10, x FOR PEER REVIEW 6 of 20
Step 7: Once we have completed Steps 3–6 and the SA condition Ti Tend is met, the process stops, and the system is considered to have frozen to the lowest temperature as defined by the Remote Sens. 2018, 10, 724 6 of 20 annealing schedule. Output the best of Si,best .
Initialize the parameters
Initialize parent population
Calculate individual membership and get corresponding fitness
Ti > Tend Y t = 0
Generate offspring through Selection, crossing, mutation
Calculate individual membership and get corresponding fitness for offspring
Simulated annealing generates new population Y N
Combining the parent population and offspring Rt, Ti+1=ρTi Sort Rt based on the nondomaination
Choose 200 to be new population
N t < Maxgen
End
Figure 2. Flow-chart of non-dominated sorting and simulated annealing genetic algorithm Figure 2. Flow-chart of non-dominated sorting and simulated annealing genetic algorithm (NSSAGA) algorithm. (NSSAGA) algorithm. 3.3. Modeling Refractivity Profile and Objective Function 3.3. Modeling Refractivity Profile and Objective Function A surface duct always forms when a relatively warm, dry air mass travels over a cooler, moister A surface duct always forms when a relatively warm, dry air mass travels over a cooler, moister air mass [26], and it may cause substantial errors in propagation calculations for low-elevation GPS air mass [26], and it may cause substantial errors in propagation calculations for low-elevation GPS signals [47]. The modified refractivity profile is commonly employed for a surface duct. The following signals [47]. The modified refractivity profile is commonly employed for a surface duct. The is a four-parameter trilinear model [48] that parameterizes the atmospheric refractivity structure following is a four-parameter trilinear model [48] that parameterizes the atmospheric refractivity (Figure3): structure (Figure 3): c z z < h 1 1 ( ) = + M z M0 c1h1 + c2c(1zz− h1) h1
Remote Sens. 2018, 10, x FOR PEER REVIEW 7 of 20
where h1 is the trapping-layer based height, c1 is the base slope, h2 is the inversion-layer thickness,Remote Sens. and2018 ,c102 , 724is the slope from h1 to h1 h2 . The slope of the top layer is 0.118 M-units/m,7 of 20 considering the layer as the standard refractivity condition.
z
CIRA+Q
h Inversion slope c2 2
Inversion slope c1 h1
N Figure 3. Geometric scheme of refractivity profile model. Figure 3. Geometric scheme of refractivity profile model.
To reconstruct the refractivity profile, a multiobjective cost function [35] (Liao et al., 2016) is usuallyTo defined reconstruct as: the refractivity profile, a multiobjective cost function [35] (Liao et al., 2016) is usually defined as: obs mod 2 obs mod 2 J1(xJ )(x)( =S (∆(xS)(x) −S∆(Sx()x) ) ) min!= min! 1 (15)(15) J (x) (P (x)obsobs P (x)modmod)2 2 min! 2 J2(x) =c (Pc(x) −c Pc(x) ) = min!
TT obsobs modmod wherewhere xx=(c (1c,1c,2c,2h,1h,1h, 2h)2) isis a a vector, vector, ∆SS((xx)) and and∆ S(xS)(x) are are the the observed observed and and simulated simulated GPS obs mod phase delays, respectively, at the same antenna height, and Pc(x) and Pcobs(x) are the observedmod and GPS phase delays, respectively, at the same antenna height, and Pc (x) and Pc (x) are the received GPS propagation losses, respectively, at the same antenna height. Here, the objective function observed and received GPS propagation losses, respectively, at the same antenna height. Here, the is the least-squares function. However, the ordinary least-squares (OLS) function cannot be minimized objective function is the least-squares function. However, the ordinary least-squares (OLS) function directly when the parameters acquire small errors in some cases. cannot be minimized directly when the parameters acquire small errors in some cases. The Bartlett objective function, which is broadly used in the field of marine acoustic tomography, The Bartlett objective function, which is broadly used in the field of marine acoustic achieves good results in estimating refractivity conditions [10]. Hence, we adopted the Bartlett power tomography, achieves good results in estimating refractivity conditions [10]. Hence, we adopted the function as the new cost function. Referring to Gerstoft et al. [10], the objective function is defined as: Bartlett power function as the new cost function. Referring to Gerstoft et al. [10], the objective function is defined as: obs mod 2 J1(x) = (∆S(x) − ∆S(x) ) = min! obs mod 2 Nx J1(x) (NS(x) S(x) 2) ∑min!PT Q dep Nelement Nx i,j,k i,j,k (16) 1 N j=1 J (x) = ∑ ∑ [ ∑ p −x ] = min! 2 Ndep Nelement i,j,k TNx k=1 i=1 j=1 P Q N 2 i,∑j,k i,ij,kj,k dep Nelement N x j=1 1 j1 (16) J 2 (x) [ pi, j,k ] min! N N N x where Pi,j,k and Qi,j,k aredep the observedelement k 11 andi calculatedj1 data, respectively,Q in dB form; they are the elements of the propagation loss matrices. Term N is the height, Ni, j,kis the range, and N is the dep j1 x element number of GPS receivers at the same antenna height. Here, the receiver heights are 20 m, 50 m, 100 m, where150 m, P andi, j,k 400and m. Q Therei, j,k are is onlythe observed one receiving and calculated antenna at data, a given respectively, height level. in dB form; they are the Section 3.2 noted that the population is sorted by the solutions sets J1 and J2 according to elements of the propagation loss matrices. Term Ndep is the height, N x is the range, and Nelement non-domination. For different antenna heights, the same values of the refractivity parameters may is producethe number a different of GPS setreceivers of solutions. at the same Thus, antenna when inferring height. Here, refractivity the receiver parameters heights based are 20 on m, data 50 m, sets 100collected m, 150 m, from and antenna 400 m. There of differing is only height, one receiving we must antenna sort the at individualsa given height according level. to whether the heightSection dependence 3.2 noted of that the the data population (phase delay is sorted and propagationby the solutions loss) sets is considered. J1 and J 2 Considering according to the non-domination.change in receiver For heightdifferent means antenna that heights, each datum the same from values an antenna of the of refractivity a given height parameters calculates may its produceown objective a different function, set of whereuponsolutions. Thus, individuals when infe associatedrring refractivity with thesame parameters height arebased sorted on data according sets collectedto their from objective antenna values of (Figurediffering4a). height, Otherwise, we must we takesort the averageindividuals objective according value to of whether the different the antenna heights and take it as the objective function of data sets with different antenna heights, sorting Remote Sens. 2018, 10, x FOR PEER REVIEW 8 of 20 height dependence of the data (phase delay and propagation loss) is considered. Considering the change in receiver height means that each datum from an antenna of a given height calculates its own objective function, whereupon individuals associated with the same height are sorted according to their objective values (Figure 4a). Otherwise, we take the average objective value of the Remote Sens. 2018, 10, 724 8 of 20 different antenna heights and take it as the objective function of data sets with different antenna heights, sorting individuals associated with the same height according to Figure 4b. Finally, the inversionindividuals method associated must withaverage the the same optimal height parameters according toSi, Figurebest from4b. the Finally, different the heights inversion as methodthe final result.must average the optimal parameters Si,best from the different heights as the final result. Herein,Herein, wewe definedefine the the Bartlett Bartlett function function as Bartlett1as Bartle iftt1 we if consider we consider the receiver the receiver height orheight as Bartlett2 or as Bartlett2if we do if not. we In do contrast not. In tocontrast the Bartlett to the function, Bartlett thefunction, OLS function,the OLS whichfunction, considers which theconsiders change the in changereceiver in height, receiver is takenheight, into is taken account into in account Section 4in as Section part of 4 the as part NSSAGA-OLS of the NSSAGA-OLS algorithm. algorithm.
J1 and J2 in 20m
J1 and J2 in Calculate the objective Models in different 50m values in the different Sortng the datas antenna height antenna height in the same （a） Refractivity structure J1 and J2 in receiver height parameters (c ,c ,h ,h ) 100m 1 2 1 2 according to J1 and J2 in objective values 150m
J1 and J2 in 400m
J1 and J2 in 20m
J1 and J2 in Models in different 50m Sortng the datas antenna height Average the objective in the same Refractivity structure J and J in values and take it as the 1 2 receiver height （b）parameters (c ,c ,h ,h ) 100m objective value in the 1 2 1 2 according to different antenna height J1 and J2 in objective value 150m
J1 and J2 in 400m
FigureFigure 4. FlowchartFlowchartof of calculating calculating objective objective value. value. (a) Consider(a) Consider the effectthe effect of antenna of antenna height. height. (b) Neglect (b) Neglectthe effect the of effect antenna of antenna height. height.
4.4. Results Results TheThe simulations assumed assumed a a GPS GPS elevation elevation angle angle of of 1°, 1◦, a a transmitting transmitting frequency frequency of of 1500 1500 MHz, MHz, andand a beambeam widthwidth ofof 1616°.◦. To To verify the feasibility and anti-noise ability, the simulated conditions containcontain 0%, 0%, 3%, 3%, 5%, 5%, 7%, 7%, or or 10% 10% Gaussian Gaussian white white noise; noise; the the 0% 0%level level is the is theideal ideal condition. condition. Here, Here, the Gaussianthe Gaussian whitewhite noise noise was added was added into the into propagation the propagation loss only loss and only the andphase the delay phase is kept delay accurate. is kept However,accurate. However,the elevation the elevationangle seriously angle seriously affects affectsthe error the errorof phase of phase delay. delay. This This uncertainty uncertainty is estimatedis estimated to be to about be about 8–10 8–10mm from mm fromthe zenith, the zenith, increasing increasing to about to 8–10 about cm 8–10 at an cm elevation at an elevation angle of 5° [49]. Therefore, noise of delay information is also necessary to consider and becomes the angle of 5◦ [49]. Therefore, noise of delay information is also necessary to consider and becomes the follow-up research. follow-up research.
4.1.4.1. Comparison Comparison between between NSSAGA NSSAGA and and NSGA-II InIn this this section, section, we we compare compare the the performance performance of of NSSAGA with with that of NSGA-II, using the same objectiveobjective function function and and the the same same receiver receiver height. height. Here Here,, the the objective objective function function is is Equation Equation (15). (15). Table Table 1 listslists thethe limitslimits of of inverting inverting parameters parameters and and the truethe true values. values. The inversion The inversion results results for different for different antenna antennaheights areheights also givenare also in Tablegiven1. in From Table Table 1. 1From, the syntheticTable 1, parametersthe synthetic retrieved parameters by NSSAGA retrieved are by NSSAGAbetter than are those better retrieved than those by the retrieved compared by algorithmthe compared NSGA-II algorithm for the NSGA-II same antenna for the height same under antenna the heightideal condition. under the Moreover, ideal condition. it is difficult Moreover, to compare it is the difficult inversion to resultscompare for the different inversion antenna results heights for differentusing the antenna same algorithm. heights using the same algorithm.
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Table 1. The value of inversion parameters under noiseless and added noise conditions Table(bold1. fontThe indicates value of best inversion retrieved parameters parameters). under noiseless and added noise conditions (bold font indicates best retrieved parameters). Parameters Inversion Slope c1 Height h1 Inversion Slope c2 Height h2 Units N-units/m m N-units/m m Parameters Inversion Slope c1 Height h1 Inversion Slope c2 Height h2 Lower bound −0.1 50 −0.4 250 Units N-units/m m N-units/m m UpperLower bound bound 0 −0.1150 50 0− 0.4 350 250 TrueUpper value bound −0.02 0100 150−0.2 0 300 350 True20 value m −0.0302 (51%)−0.02 107.4749 (7.5%) 100 −0.1979 −(−0.21.1%) 296.8430 300 (−1.1%) 100 m20 m −0.0524−0.0302 (162%) (51%) 116.8249107.4749 (16.8%) (7.5%) −0.1894−0.1979 (−5.3%) (−1.1%) 295.6861 296.8430 ((−−1.4%)1.1%) NSSAGA 100 m −0.0524 (162%) 116.8249 (16.8%) −0.1894 (−5.3%) 295.6861 (−1.4%) NSSAGA 150 m150 m −0.0264−0.0264 (32%) (32%) 104.6840 104.6840 (4.7%) (4.7%) −0.1952−0.1952 (−2.4%) (−2.4%) 305.6387 305.6387 (1.9%) 400 m400 m −0.0414−0.0414 (107%) (107%) 106.5472 106.5472 (6.6%) (6.6%) −0.1966−0.1966 (−1.7%) (−1.7%) 286.0859 286.0859 ((−−4.6%)4.6%) 20 m20 m −0.0247−0.0247 (23.5%) (23.5%) 51.384451.3844 (−48.6%) (−48.6%) −0.2103−0.2103 (5.2%) (5.2%) 288.6211 288.6211 (−3.8%) 100 m −0.0615 (207.5%) 83.1727 (−16.8%) −0.2213 (10.7%) 255.8522 (−14.7%) NSGA-II 100 m −0.0615 (207.5%) 83.1727 (−16.8%) −0.2213 (10.7%) 255.8522 (−14.7%) NSGA-II 150 m −0.0767 (283.5%) 111.1971 (11.2%) −0.1633 (−18.4%) 326.9079 (9.0%) 150 m400 m−0.0767−0.0021 (283.5%) (−89.5%) 111.1971 77.9154 (11.2%) (−22.1%) −0.1633−0.2430 (−18.4%) (21.5%) 326.9079 253.9479 ( −(9.0%)15.4%) 400 m −0.0021 (−89.5%) 77.9154 (−22.1%) −0.2430 (21.5%) 253.9479 (−15.4%) Figure5 illustrates the retrieved refractivity profile and the corresponding absolute error. Figure 5 illustrates the retrieved refractivity profile and the corresponding absolute error. The The numbers in the legends represent the antenna height. Because the discrepancy between the numbers in the legends represent the antenna height. Because the discrepancy between the retrieved and simulated profiles is large for the antenna height of 50 m, Figure5 gives the retrieved retrieved and simulated profiles is large for the antenna height of 50 m, Figure 5 gives the retrieved profile excluding its results. Obviously, the retrieved profile based on NSSAGA for the same antenna profile excluding its results. Obviously, the retrieved profile based on NSSAGA for the same height is closer to the real profile than those based on NSGA-II. The absolute error also shows that antenna height is closer to the real profile than those based on NSGA-II. The absolute error also NSSAGAshows that performs NSSAGA better performs than NSGA-II.better than The NSGA-II. error of The NSSAGA_20 error of NSSAGA_20 versus height versus is less height than is 1 less N-unit, whilethan the1 N-unit, error ofwhile NSSAGA_150 the error of is NSSAGA_150 less than that is of le thess otherthan that algorithms of the other except algorithms NSSAGA_20. except Thus, theNSSAGA_20. best retrieval Thus, results the arebest achieved retrieval whenresults the are receiver achieved is locatedwhen the inside receiver the duct.is located In general, inside thethe new algorithmduct. In general, NSSAGA the outperformsnew algorithm NSGA-II NSSAGA in multiobjectiveoutperforms NSGA-II inversion. in multiobjective In other words, inversion. adding theIn SA algorithmother words, has adding improved the SA NSGA-II. algorithm has improved NSGA-II.
Figure 5. Retrieved refractivity profiles and corresponding absolute error by using (a,b) NSSAGA Figure 5. Retrieved refractivity profiles and corresponding absolute error by using (a,b) NSSAGA and and (c,d) non-dominated sorting genetic algorithm II (NSGA-II) with antenna heights of 20 m, 100 m, (c,d) non-dominated sorting genetic algorithm II (NSGA-II) with antenna heights of 20 m, 100 m, 150 m, 150 m, and 400 m. and 400 m.
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4.2.4.2. NSSAGA NSSAGA with with Different Different Levels Levels of of Gaussian Gaussian Noise Noise MostMost practical practical applications applications of ofGPS GPS observations observations involve involve measurement measurement errors. errors. In this In section, this section, we presentwe present retrievalretrieval simulations simulations that that involve involve measur measurementement errors errors that that we we assume assume obey obey Gaussian Gaussian statistics.statistics. The The detailed detailed retrieval retrieval results results are are presente presentedd in in Figure Figure 66 andand Table Table 22,, wherewhere thethe blackblack lineline inin FigureFigure 66 isis the the true true refractivity refractivity profile. profile. The The inverted inverted refractivity refractivity profiles profiles marked marked by by the the red red solid solid line, line, thethe blue blue dash dash line, line, and and the the dash-dot dash-dot or or dotted dotted lines lines of of various various colors colors correspond correspond to to 0%, 0%, 3%, 3%, 5%, 5%, 7%, 7%, andand 10% 10% Gaussian Gaussian noise, noise, respectively. respectively. To To make make the the comparison comparison more more intuitive, intuitive, we we used used bold bold font font for for thethe best best inferred inferred refractivity refractivity parameters. parameters. It It is is ob obviousvious that that NSSAGA NSSAGA gives gives the the best best inversion inversion results results whenwhen the the antenna antenna height height is is 20 20 m. m. The The hybrid hybrid algo algorithmrithm performs performs worst worst when when the the antenna antenna height height is is 5050 m. m. These These results results are are also also shown shown in in Figure Figure 6.6. FigureFigure 6 shows that noise degrades the retrieval, especially for for the the antenna height height of of 50 50 m, m, whichwhich is is why why thosethose data data were were excluded excluded from from the the comparison comparison in in Section Section 4.1.4.1. For For the the 20 20 m m antenna antenna height,height, the the simulation simulation without without noise noise gives gives the the best best re results.sults. The The retrieval retrieval errors errors are are similar similar for for the the 3%, 3%, 5%,5%, and and 7% 7% noise noise conditions, conditions, with with the the discrepancie discrepanciess not not exceeding exceeding 15 15 N-units. N-units. Besides Besides the the antenna antenna heightheight of of 20 20 m, m, the the error error for for 150 150 m m is is also also within within 20 20 N-units N-units when when adding adding no no more more than than 7% 7% Gaussian Gaussian noise.noise. The The retrievals retrievals for for the the 100 100 m m and and 400 400 m m antenn antennaa heights heights are are much much better better than than that that for for 50 50 m m but but poorerpoorer that thosethose forfor 20 20 m m and and 150 150 m. Thus,m. Thus, the hybrid the hybrid algorithm algorithm has strong has noisestrong immunity noise immunity according accordingto its performance to its performance with different with levelsdifferent of Gaussian levels of noise.Gaussian noise.
Height=20m
Figure 6. Cont.
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Height=50m
Height=100m
Height=150m
Figure 6. Cont.
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Height=400m
FigureFigure 6. 6. TheThe retrieved retrieved refractivity refractivity profile profile and and corre correspondingsponding absolute absolute error error by by NSSAGA NSSAGA under under differentdifferent Gaussian Gaussian Noise Noise levels levels.. Antenna Antenna heights heights used are ((aa,,bb)) 2020 m;m; ((cc,d,d)) 50 50 m; m; ( e(e,f,)f) 100 100 m; m;(g (,gh,h) 150) 150 m ; m;(i ,(ji),j 400) 400 m. m.
TableTable 2. 2. The valuevalue of of inversion inversion parameters parameters under under different different antenna antenna heights andheights Gaussian and Gaussian noise levels noise(bold levels font indicates (bold font best indicates retrieved parameters).best retrieved parameters). Antenna Height and Antenna Height and Inversion Slope c1 Height h1 Inversion Slope c2 Height h2 Gaussian Noise Level Inversion Slope c Height h Inversion Slope c Height h Gaussian Noise Level 1 1 2 2 3% −0.0345 (72.5%) 132.8208 (32.8%) −0.2367 (18.4%) 302.0072 (0.7%) 5% 3% −0.0366−0.0345 (83%) (72.5%) 113.3292 132.8208 (13.3%) (32.8%) −−0.24150.2367 (20.8%) (18.4%) 302.0072 290.9717 (0.7%)(−3%) 20 m 5% −0.0366 (83%) 113.3292 (13.3%) −0.2415 (20.8%) 290.9717 (−3%) 20 m 7% −0.0274 (37%) 106.8854 (6.9%) −0.2338 (16.9%) 310.0565 (3.4%) 10% 7% −0.0504−0.0274 (152%) (37%) 127.7177106.8854 (27.7%) (6.9%) −−0.28160.2338 (40.8%) (16.9%) 310.0565318.0503 (3.4%)(6%) 10% −0.0504 (152%) 127.7177 (27.7%) −0.2816 (40.8%) 318.0503 (6%) 3% −0.0419 (109.5%) 76.6365 (−23.4%) −0.2886 (44.3%) 279.2205 (−6.9%) 5% 3% −0.0650−0.0419 (225%) (109.5%) 63.5319 76.6365 (−36.5%) (−23.4%) −−0.32990.2886 (65.0%) (44.3%) 279.2205265.9958 ((11.3%)−6.9%) 50 m 5% −0.0650 (225%) 63.5319 (−36.5%) −0.3299 (65.0%) 265.9958 (11.3%) 50 m 7% −0.0476 (138%) 93.6565 (−6.3%) −0.3476 (73.8%) 281.2179 (−6.3%) 10% 7% −0.0737−0.0476 (268.5%) (138%) 59.122293.6565 (−40.9%) (−6.3%) −−0.34760.39 (95%) (73.8%) 281.2179284.2965 (−5.2%)6.3%) − − − − 3% 10% −0.05970.0737 (198.5%) (268.5%) 113.8418 59.1222 (13.8%) ( 40.9%) −0.22890.39 (14.5%) (95%) 284.2965 302.8125 ((0.9%)5.2%) 5% 3% −0.0372−0.0597 (86%) (198.5%) 120.9352 113.8418 (20.9%) (13.8%) −−0.23640.2289 (18.2%) (14.5%) 299.3216 302.8125 (− (0.9%)0.2%) 100 m 7% 5% −0.0565−0.0372 (182.5%) (86%) 132.5776 120.9352 (32.6%) (20.9%) −−0.25760.2364 (28.8%) (18.2%) 299.3216 311.2225 ( −(3.7%)0.2%) 100 m 10% 7% −0.0344−0.0565 (72%) (182.5%) 136.8554 132.5776 (36.9%) (32.6%) −0.3036−0.2576 (−51.8%) (28.8%) 316.0985 311.2225 (5.4%) (3.7%) − − − 3% 10% −0.02130.0344 (6.50%) (72%) 120.1325136.8554 (20.1%) (36.9%) −0.23550.3036 (17.8%) ( 51.8%) 307.1159316.0985 (2.4%) (5.4%) 5% 3% −0.0402−0.0213 (101%) (6.50%) 102.1663120.1325 (2.2%) (20.1%) −0.25400.2355 (27%) (17.8%) 295.8991 307.1159 (− (2.4%)1.4%) 150 m 7% 5% −0.0337−0.0402 (68.5%) (101%) 92.4479102.1663 (−7.6%) (2.2%) −0.2503−0.2540 (25.2%) (27%) 295.8991279.4482 (−6.9%)1.4%) 150 m 10% 7% −0.0487−0.0337 (143.5%) (68.5%) 104.5684 92.4479 (4.6%) (−7.6%) −−0.30240.2503 (51.2%) (25.2%) 279.4482 313.3066 ((4.4%)−6.9%) 3% 10% −0.0350−0.0487 (75%) (143.5%) 106.1164104.5684 (6.1%) (4.6%) −−0.22180.3024 (11.0%) (51.2%) 315.5805 313.3066 (5.2%) (4.4%) 5% 3% −0.0476−0.0350 (138%) (75%) 91.6220106.1164 (−8.4%) (6.1%) −−0.27010.2218 (35.1%) (11.0%) 286.7822315.5805 (− (5.2%)4.4%) 400 m 7% 5% −0.03−0.0476 (50%) (138%) 90.6699 91.6220 (−9.3%) (−8.4%) −−0.29990.2701 (50.0%) (35.1%) 286.7822293.5933 (−2.1%)4.4%) 400 m 10% 7% −0.0774−0.03 (287%) (50%) 126.2492 90.6699 (26.3%) (−9.3%) −−0.29690.2999 (48.5%) (50.0%) 293.5933298.8726 (−0.4%)2.1%) 10% −0.0774 (287%) 126.2492 (26.3%) −0.2969 (48.5%) 298.8726 (−0.4%) 4.3. Analysis of Retrieved Propagation Loss 4.3. Analysis of Retrieved Propagation Loss Figure 7a,c depict the propagation loss of the GPS signal over horizontal distances of 0–200 km at a heightFigure of7 a,c100 depictm for thedifferent propagation antenna loss heights, of the showing GPS signal that over the horizontalpropagation distances loss decreases of 0–200 with km rangeat aheight in each of 100case. m Obviously, for different the antenna overall heights, trend showingof the propagation that the propagation loss remains loss decreasesthe same withfor differentrange in antenna each case. heights. Obviously, The amplitude the overall trendof the ofpropagation the propagation loss does loss remainsnot change, the same but changes for different in interferenceantenna heights. pattern The alter amplitude its position. of the Those propagation positional loss changes does not demonstrate change, but that changes propagation in interference losses forpattern different alter antenna its position. heights Those could positional be combined changes to re demonstratetrieve the refractivity that propagation profile lossesmore effectively. for different Basedantenna on this heights feature, could we be integrated combined the to measurem retrieve theents refractivity from different profile receivers more effectively.and constructed Based the on Bartlett function to retrieve the atmospheric refractivity structure. Figure 7b,d show how the propagation loss of the GPS signal changes with height at a distance of 100 km. The amplitude of the
Remote Sens. 2018, 10, 724 13 of 20 this feature, we integrated the measurements from different receivers and constructed the Bartlett function to retrieve the atmospheric refractivity structure. Figure7b,d show how the propagation loss of the GPS signal changes with height at a distance of 100 km. The amplitude of the propagation loss in the ductRemote layerSens. 2018 remains, 10, x FORaround PEER REVIEW 100 dB, indicating vividly the trapping effect of the 13 surface of 20 duct. Remote Sens. 2018, 10, x FOR PEER REVIEW 13 of 20 As Figurepropagation7 shows, loss the in retrieved the duct andlayer simulatedremains around propagation 100 dB, indicating losses differ vividly little. the trapping effect of propagationFigurethe surface8 shows loss duct. in the the As absolute duct Figure layer 7 errorshows, remains between the around retrieved the 100 and inversion dB, simulated indicating results propagation vividly and the the lossestrapping simulated differ effect little. true of value. It is foundthe thatsurface NSSAGAFigure duct. 8 As shows keepsFigure the 7 the absoluteshows, absolute the error retrie error betweenved within and the simulated inve modestrsion propagation results bounds. and Because thelosses simulated differ of the little. true model’s value. defects,It a larger errorisFigure found arises 8 showsthat beyond NSSAGA the absolute 100 keeps km, error butthe between absolute the inverted the error inve profilewithinrsion results modest can stilland bounds. depictthe simulated Because the refractivity true of value.the model’s It environment roughly.is founddefects, Overall, that a largerNSSAGA based error on keeps arises the retrievedthe beyond absolute 100 propagation errorkm, but within the loss, invertedmodest NSSAGA profilebounds. can is Because highly still depict of capable the the model’s refractivity of retrieving the environment roughly. Overall, based on the retrieved propagation loss, NSSAGA is highly capable refractivitydefects, a structurelarger error at arises distances beyond up 100 to km, 100 but km. the inverted profile can still depict the refractivity environmentof retrieving roughly. the refractivity Overall, based structure on the at distancesretrieved uppropagation to 100 km. loss, NSSAGA is highly capable of retrieving the refractivity structure at distances up to 100 km.
Figure 7. The propagation loss of ground-based GPS signal. (a,c) Simulation and inversion results at Figure 7. The propagation loss of ground-based GPS signal. (a,c) Simulation and inversion results at Figuredistance 7. The ofpropagation 0–200 km andloss ofheight ground-based of 100 m; GPS(b,d) signal. simulation (a,c) Simulationand inversion and results inversion at distance results atof 100 distance of 0–200 km and height of 100 m; (b,d) simulation and inversion results at distance of 100 km distancekm andof 0–200 height km of and0–400 height m. of 100 m; (b,d) simulation and inversion results at distance of 100 andkm height and height of 0–400 of 0–400 m. m.
Figure 8. The absolute error of propagation loss. (a) Distance of 0–200 km and height of 100 m and (b) distance of 100 km and height of 0–400 m. FigureFigure 8. 8.The Theabsolute absolute error error of ofpropagation propagation loss. ( loss.a) Distance (a) Distance of 0–200 km of and 0–200 height km of and 100 heightm and (b of) 100 m and distance of 100 km and height of 0–400 m. (b) distance of 100 km and height of 0–400 m.
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4.4.Remote NSSAGA Sens. 2018 with, 10, Differentx FOR PEER Objective REVIEW Functions 14 of 20
4.4.Next, NSSAGA we integratedwith Different all Objective the data Functions from antenna of different heights and used them to retrieve the refractivity structure. Figure9 shows the distinctions among the different objective functions, Next, we integrated all the data from antenna of different heights and used them to retrieve the and the objective functions of Equations (14) and (15) are compared in Table3. From Table3, it is easily refractivity structure. Figure 9 shows the distinctions among the different objective functions, and seen that OLS offers the best performance in retrieving refractivity parameters. However, this good the objective functions of Equations (14) and (15) are compared in Table 3. From Table 3, it is easily parametric inversion does not mean that the inversion profiles are closer to the true profiles. seen that OLS offers the best performance in retrieving refractivity parameters. However, this good Figure9 illustrates the results obtained with the Bartlett objective function and the least-squares parametric inversion does not mean that the inversion profiles are closer to the true profiles. function.Figure Bartlett1 9 illustrates (the red the solidresults line) obtained takes with the antennathe Bartlett height objective information function and into the account, least-squares whereas Bartlett2function. (the Bartlett1 blue dashed (the red line) solid does line) not. takes OLS the provides antennainversion height information results from into all account, data based whereas on the least-squaresBartlett2 (the method blue dashed considering line) does the not. antenna OLS prov heightides information. inversion results In comparison from all data with based Antenna on the 20 andleast-squares Antenna150 method, Bartlett1 considering provides the fairly antenna good height estimates information. and is even In comparison better than with Bartlett2. Antenna Here,20 Antennaand Antenna20 and150 Antenna, Bartlett1150 providesimply the fairly reconstructed good estimates results and from is even the receivedbetter than information Bartlett2. Here, at 20 m andAntenna 150 m.20 Becauseand Antenna the lots150 imply of detection the reconstructed information results could from improve the received the accuracy information of inversion, at 20 m OLS isand expected 150 m. to Because perform the better lots of thandetection Antenna information20 and Antennacould improve150. The the discrepancyaccuracy of inversion, in absolute OLS error is betweenexpected Bartlett1 to perform and OLS better is less than than Antenna 1 N-unit,20 and meaning Antenna that150 Bartlett1. The discrepancy is an acceptable in absolute alternative errorfor refractivitybetween Bartlett1 inversion. and OLS is less than 1 N-unit, meaning that Bartlett1 is an acceptable alternative for refractivity inversion. Table 3. The value of inversion parameters by NSSAGA with different objective functions (bold font indicatesTable 3. best The retrieved value of parameters). inversion parameters by NSSAGA with different objective functions (bold font indicates best retrieved parameters).
Objective Function Inversion Slope c1 Height h1 Inversion Slope c2 Height h2 Objective Function Inversion Slope c1 Height h1 Inversion Slope c2 Height h2 Bartlett1Bartlett1 −−0.03060.0306 (53%)(53%) 104.3187 104.3187 (4.3%) (4.3%) −0.2024−0.2024 (1.2%) (1.2%) 289.7689 289.7689 (−3.4%) (−3.4%) Bartlett2Bartlett2 −−0.05050.0505 (152.5%)(152.5%) 124.6875 124.6875 (24.7%) (24.7%) −0.1998−0.1998 (−0.1%) (− 0.1%)279.5246279.5246 (−6.8%) (−6.8%) NSSAGANSSAGA OLSOLS −0.0158−0.0158 ( −(−21%)21%) 101.5392 101.5392 (1.5%) (1.5%) −0.2018−0.2018 (0.9%) (0.9%) 301.2038301.2038 (0.4%) (0.4%) 20 m20 m −−0.03020.0302 (51%)(51%) 107.4749 107.4749 (7.5%) (7.5%) −0.1979−0.1979 (−1.1%) (− 1.1%)296.8430 296.8430 (−1.1%) (−1.1%) 150 m150 m −−0.02640.0264 (32%)(32%) 104.6840 104.6840 (4.7%) (4.7%) −0.1952−0.1952 (−2.4%) (− 2.4%)305.6387 305.6387 (1.9%) (1.9%)
Figure 9. Comparison of retrieved refractivity profile and absolute error by NSSAGA with various Figure 9. Comparison of retrieved refractivity profile and absolute error by NSSAGA with various objective functions. objective functions. To assess the performance of NSSAGA with three objective functions under different Gaussian noiseTo assessconditions, the performancethe optimal refractivity of NSSAGA retrievals with three are provided objective in functions Figure 10 under and Table different 4. Using Gaussian the noiseBartlett conditions, objective the function optimal improves refractivity the retrievalsinversion areability provided of NSSAGA in Figure under 10 and Gaussian Table4 .noise Using theconditions, Bartlett objective and Bartlett1 function is more improves accurate the than inversion Bartlett2 ability below of 100 NSSAGA m. The underlimited Gaussian simulations noise conditions,presented andherein Bartlett1 suggest is that more Bartlett1 accurate provides than Bartlett2better inversion below 100results m. and The has limited stronger simulations noise immunity for refractivity estimation.
Remote Sens. 2018, 10, 724 15 of 20 presented herein suggest that Bartlett1 provides better inversion results and has stronger noise immunity for refractivity estimation. Remote Sens. 2018, 10, x FOR PEER REVIEW 15 of 20
FigureFigure 10. The 10. retrievedThe retrieved refractivity refractivity profileprofile and corre correspondingsponding absolute absolute error errorby NSSAGA by NSSAGA with with differentdifferent objective objective functions. functions. (a,b) The(a,b) BartlettThe Bartlett objective objective function function with with the receiver the receiver height height information; (c,d) the Bartlett objective function without the receiver height information; (e,f) the least-squares method with the receiver height information. Remote Sens. 2018, 10, 724 16 of 20
Table 4. The value of inversion parameters by NSSAGA with different objective functions and Gaussian noise (bold font indicates best retrieved parameters).
Objective Function Inversion Slope c Height h Inversion Slope c Height h and Noise Level 1 1 2 2 3% −0.0476 (138%) 125.2315 (25.2%) −0.2287 (14.4%) 306.4400 (2.2%) 5% −0.0578 (189%) 132.5299 (32.5%) −0.2484 (24.2%) 312.4563 (4.2%) Bartlett1 7% −0.0557 (178.5%) 117.3913 (17.4%) −0.2791 (39.6%) 321.0724 (7.0%) 10% −0.0572 (186%) 125.0450 (25.1%) −0.2768 (38.4%) 339.8595 (13.3%) 3% −0.0461 (130.5%) 119.1744 (19.2%) −0.2043 (2.2%) 311.4039 (3.8%) 5% −0.0497 (148.5%) 123.1673 (23.2%) −0.2330 (16.5%) 308.2942 (2.8%) Bartlett2 7% −0.0453 (126.5%) 138.2198 (38.2%) −0.2799 (40.0%) 308.1836 (2.7%) 10% −0.0470 (135%) 109.1366 (9.1%) −0.2911 (45.6%) 327.2096 (9.1%) 3% −0.0408 (104%) 123.0645 (23.1%) −0.2418 (20.9%) 304.0172 (1.3%) 5% −0.0599 (199.5%) 140.5727 (40.6%) −0.2423 (21.2%) 329.4880 (9.8%) OLS 7% −0.0498 (149%) 111.2574 (11.3%) −0.2934 (46.7%) 304.5782 (1.5%) 10% −0.0523 (161.5%) 125.7983 (25.8%) −0.3477 (73.9%) 312.7544 (4.3%)
4.5. Experimental Data Testing To demonstrate that the NSSAGA can provide sufficient retrieval accuracy under realistic conditions, the excess phase and propagation loss measured by ground-based GPS receiver were used. The excess phase was estimated using a modified version of the Bernese v5.0 software, and the propagation loss was calculated according to the receiving power density of the antenna [35]. Zero-differenced observations were processed in precise point positioning module and the tropospheric zenith delay (phase delay) module produces cleaned post-fit residuals and forms baselines. Finally, slant tropospheric delay was determined through parameter estimation [50]. The corresponding sounding data were obtained from meteorological-rocket detection data (~4 m sampling at altitudes less than 1.5 km) and low-resolution radiosondes (~50 m sampling at altitudes less than 6 km). These experimental data were collected from observations made in the vicinity of Poyang Lake, China (115.27◦E, 29.11◦N). Table5 gives the inversion parameters obtained using different inversion methods, and Figure 11 shows the differences between the estimated results and the sounding data. Here, only two groups of ground-based GPS data were retrieved (Figure 11a,b: the observed time is 06:17, and Figure 11c,d: the observed time is 18:42). In Figure 11, the estimated profiles along with the measured profiles are on the left side of the figure. Although the real data contain some observation errors, the estimated profiles are consistent with the measured profiles in general. In these observed cases, Bartlett1 and Bartlett2 provide better fits. However, the absolute error demonstrates that the refractivity profiles estimated using Bartlett1 are much closer to the real refractivity environment than is the case for Bartlett2 or OLS. Also, from the root-mean-square errors of the different inversion methods in the two cases, we conclude that Bartlett1 can be applied to realistic conditions with sufficient retrieval accuracy.
Table 5. The value of inversion parameters by different inversion methods.
Inversion Slope Inversion Slope RMS Error Parameters Height h1 (m) Height h2 (m) c1 (N-Units/m) c2 (N-Units/m) (N) Bartlett1 −0.0746 110.9846 −0.1960 317.8961 11.4734 Case1 Bartlett2 −0.0867 127.4610 −0.1875 303.5692 12.6298 OLS −0.0627 94.7159 −0.2120 314.6175 14.3666 Bartlett1 −0.1689 77.0123 −0.2138 341.3519 8.9907 Case2 Bartlett2 −0.1477 76.1761 −0.2549 351.2348 9.0134 OLS −0.1619 87.1056 −0.2513 342.0737 9.2483 Remote Sens. 2018, 10, 724 17 of 20 Remote Sens. 2018, 10, x FOR PEER REVIEW 17 of 20
FigureFigure 11. 11.Refractivity Refractivity profiles profiles byby differentdifferent inversioninversion methods for for real real atmospheric atmospheric environment. environment. (a,b(a):,b time): time for for Case Case 1 is1 is 06:17; 06:17; (c (,cd,d):): time time for for CaseCase 22 isis 18:42.18:42.
Table 5. The value of inversion parameters by different inversion methods. 5. Conclusions Inversion Slope c1 Inversion Slope c2 RMS Parameters Height h1 (m) Height h2 (m) A surface duct can substantially(N-Units/m) affect the performance(N-Units/m) of radar or communicationError (N) systems designed accordingBartlett1 to standard−0.0746 atmospheric 110.9846 conditions. Hence,−0.1960 estimating 317.8961 refractivity profiles11.4734 plays an importantCase1 roleBartlett2 in predicting −0.0867 the performance127.4610 of electromagnetic −0.1875 systems [26303.5692]. In this paper,12.6298based on OLS −0.0627 94.7159 −0.2120 314.6175 14.3666 previous workBartlett1 [35], we put−0.1689 forward an improved77.0123 retrieval−0.2138 method, using341.3519 the hybrid8.9907 NSSAGA algorithmCase2 with Bartlett2 match-field processing−0.1477 to retrieve76.1761 tropospheric−0.2549 refractivity profiles.351.2348 The present9.0134 study made use of ground-basedOLS GPS−0.1619 phase delay87.1056 and propagation −0.2513 loss from antenna342.0737 of multiple9.2483 heights to estimate the surface duct. In the simulations, conventional NSGA-II was chosen to test the performance 5. Conclusions of NSSAGA under different levels of noise. The presented results demonstrated that NSSAGA is more accurateA andsurface that duct the invertingcan substantially profiles affect for the the hybrid performance algorithm of radar are much or communication closer to the simulatedsystems profiledesigned than according those for to NSGA-II standard when atmospheric using theconditio samens. objective Hence, estimating function. re Underfractivity Gaussian profiles noise plays of differentan important levels, role the estimatedin predicting results the performance demonstrated of thatelectromagnetic NSSAGA has systems strong [26]. noise In this immunity paper, andbased can beon applied previous to practical work [35], situations. we put forward an improved retrieval method, using the hybrid NSSAGA algorithmFurthermore, with match-field in comparing processing two objective to retrieve functions, tropospheric the match-field refractivity processing profiles. The method present gave resultsstudy that made were use better of ground-based than those of GPS the phase least-squares delay and method propagation in simulations. loss from The antenna refractivity-profile of multiple heights to estimate the surface duct. In the simulations, conventional NSGA-II was chosen to test the estimates by Bartlett1 under Gaussian noise of different levels were very good, whereas those by other performance of NSSAGA under different levels of noise. The presented results demonstrated that inversion methods were less accurate. The simulations demonstrated that Bartlett1 is an acceptable NSSAGA is more accurate and that the inverting profiles for the hybrid algorithm are much closer to alternative for refractivity inversion. Actual measurements were also used to test the improved the simulated profile than those for NSGA-II when using the same objective function. Under inversion method, and the retrieved results showed that the new method can depict the real refractivity Gaussian noise of different levels, the estimated results demonstrated that NSSAGA has strong noise environment.immunity and However, can be applie thed effect to practical of elevation situations. angle on the improved inversion method was not consideredFurthermore, herein and in willcomparing be the subjecttwo objective of future functi investigations.ons, the match-field processing method gave results that were better than those of the least-squares method in simulations. The Author Contributions: Z.S. and H.S. conceived and designed the experiments; J.X. performed the experiments; Q.L.refractivity-profile and H.Y. analyzed estimates the data; Q.L. by Bartlett1 wrote the under paper. Gaussian noise of different levels were very good, whereas those by other inversion methods were less accurate. The simulations demonstrated that Acknowledgments: This research was supported by National Natural Science Foundation of China (Grants 41375028,Bartlett1 41475021) is an acceptable and Natural alternative Science Foundation for refractivi ofty Jiangsu, inversion. China Actual (Grant measurements BK20151446). were also used to test the improved inversion method, and the retrieved results showed that the new method can Conflicts of Interest: The authors declare no conflict of interest.
Remote Sens. 2018, 10, 724 18 of 20
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