Assessment of Soil Parameter Estimation Errors for Fusion of Multichannel Radar Measurements

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1 Assessmet of Soil Parameter Estimatio Errors for Fusio of Multichael Radar Measuremets A. urei, D. Marshall, D. Radford,. Lever Cardiff Uiversity 5 The Parade, Roath, CF4 0YF, Cardiff, U A.urei@cs.cf.ac.u dave@cs.cf.ac.u D.L.J.Radford@cs.cf.ac.u Lever@cf.ac.u G. ulemi Istitute of Radiophysics ad Electroics Natioal Academy of Scieces of Uraie 1 Proscura St., harov, Uraie gulemi@ire.harov.ua Abstract The applicatio of multichael radar measuremet techiques for estimatio of bare soil parameters is based o differet priciples of radiowave ad soil surface iteractio depedig o radiowave frequecy, polarisatio ad icidece agle. The accuracy of soil parameter estimatio depeds o the umber of radar measuremets ad the choice of radiowave parameters. Radom ad systematic errors preset i radar data may also have the impact o estimatio results. To improve the accuracy of soil parameters estimatio by fusio of multichael radar data we propose a ew method for assessmet of estimatio errors. It is based o local liear approximatio of the radiowave scatterig model ad taes ito accout impairmet characteristics, measuremet coditios ad radar parameters. This ew method is applied to a example to illustrate how the estimatio accuracy of soil moisture ad roughess parameters ca be improved by optimisig the radar operatig frequecies. eywords: Data fusio, errors, radar measuremets, radar scatterig model, soil roughess, soil moisture. 1 Itroductio Multichael measuremets are frequetly used i radar remote sesig of the Earth s surface to improve the iformatio cotet of radar data, to icrease the accuracy ad reliability of measuremet results ad to estimate several parameters of the observed surface objects simultaeously. The high efficiecy of the multichael approach i retrievig the desired physical parameters of the Earth s surface is provided by exploitig the differet ways radiowaves iteract with objects depedig o frequecy, polarisatio ad icidece agle of the scattered sigal [1]. The itesity of the radar sigal scattered from the Earth s surface depeds o surface geometric structure ad dielectric characteristics. The form of this depedece is also a fuctio of radar sigal parameters, which ca be modified to improve the iformatio cotet of the scattered sigal. Radar measuremets of the Earth s surface at differet radiowave frequecies, polarisatios ad icidece agles obtaied simultaeously or withi short time itervals provide detailed iformatio about sesed objects ad improve the accuracy of iformatio retrieval. The efficiecy of multichael radar measuremets depeds o the choice of radar sigal parameters, the umber of measuremets ad the impact of measuremet errors []. Hece, the multichael techique ca achieve high accuracy by optimisatio of radar parameters ad miimisatio of radar measuremet errors. A importat applicatio of multichael radar methods for soil remote sesig is the estimatio of moisture ad surface roughess parameters [3]. It has bee show i [,4] that it is theoretically possible to estimate the roughess ad moisture characteristics of soil with high accuracy by usig multifrequecy ad dual-polarisatio sigals whe radiowave frequecies ad icidece agles of measuremets are selected properly. The drawbacs of radar data fusio based o theoretical models are elimiated by applicatio of empirical radar cross sectio (RCS models based o experimetal measuremets [5]. The algorithm proposed by Oh et al. i [5] for simultaeous estimatio of soil moisture ad root mea square height parameters is based o fusio of three radar measuremets at vertical (VV, horizotal (HH orthogoal polarisatios ad cross-polarisatio compoet (HV. The radar measuremets at differet polarisatios are fused by iversio of the depedecies betwee the scatterig sigal ad soil parameters. The depedecies are derived from the empirical soil scatterig model developed by Oh et al. [5]. This approach has bee exteded i [6], where more tha three radar measuremets at several frequecies ad icidece agles are fused to improve the accuracy ad reliability of soil parameter estimates. The algorithm [6] is based o miimisatio of the discrepacies betwee the measured radar sigal scattered by soil surface ad the sigal predicted by a empirical or theoretical RCS model. While the accuracy of soil parameter estimatio is improved by icreasig the umber of idepedet radar measuremets at differet frequecies ad icidece agles, the degree of improvemet also depeds o radar parameters. The stadard approach for selectig optimal frequecy ad icidece agle radar parameters assumes the provisio of high sesitivity of radar measuremets with respect to variatios of soil parameters []. For applicatios dealig with simultaeous estimatio of several parameters the optimisatio of radar data fusio performace requires joit aalysis of differet factors

2 which have a complicated ifluece o the sesitivity of radar measuremets. High sesitivity of measuremets should be provided simultaeously for several estimated parameters ad applicatio of the stadard approach for radar parameter optimisatio is complicated. The ew method for assessmet of soil parameter estimatio errors proposed i this paper ca be applied for ay umber of estimated soil parameters ad for fusio of radar measuremets with differet characteristics. The method is based o liear approximatio of the soil scatterig model ad uses the criterio of mea square error betwee the estimated ad true soil parameters for evaluatio of measuremet errors. It taes ito accout both radom ad systematic measuremet errors ad ca be used for optimisatio of the efficiecy of multichael radar measuremets i combiatio with ay theoretical ad empirical radar scatterig models for bare soil surfaces. The applicatio of the proposed method for optimisatio of radar measuremets is cosidered for the empirical RCS model of soil surface proposed by Oh et al. [5]. The paper is orgaised as follows. I sectio we cosider differet factors that affect the multichael radar data fusio results ad soil parameter estimatio results. The statistical characteristics of data fusio errors are derived for the ratios of RCS measuremets at VV, HH ad HV polarisatios. The method for evaluatig multichael data fusio ad soil parameter estimatio errors is preseted i Sectio 3. The experimetal results of applyig the proposed method to the assessmet of two-frequecy multipolarisatio data fusio ad to the optimisatio of radar parameters are preseted i sectio 4. Fial coclusios are give i Sectio 5. Soil Roughess ad Moisture Cotet Estimatio Errors For multichael radar remote sesig techiques the accuracy of soil parameter estimatio depeds o a umber of factors. Some of them ca be eglected while the ifluece of the others is sigificat ad has to be tae ito accout. These factors are: - RCS measuremet errors at differet frequecies ad polarisatios; - the discrepacies betwee measured ad predicted itesities of the scattered sigal due to iaccuracies of the radar scatterig model; - the depedecies betwee scattered sigal itesity ad measured soil parameters (volumetric moisture cotet m v, root-mea-square (rms height of surface roughess σ h ad soil compositio; - radar measuremet parameters, such as the umber of RCS measuremets, operatig frequecies ad icidece agles; - systematic errors itroduced by the scatterig model, radar sesor ad data fusio algorithm. RCS measuremet errors usually cotai both radom ad systematic compoets which ca be cosidered as fluctuatig ad bias error sigals respectively. The fluctuatig compoet is caused by equipmet iteral oise ad the iterferece ature of the scattered sigal. Iteral equipmet oise has the properties of additive white Gaussia oise with zero mea ad variace σ i. The itesity of iteral oise depeds o the characteristics of the radar receiver ad has fixed level selected at the stage of radar system desig. The fluctuatios i the radar sigal due to iterferece are cosidered as statistically idepedet multiplicative oise with a itesity that is usually much higher tha the itesity of iteral equipmet oise [1]. Radar sigal fluctuatios are suppressed by averagig of idepedet radar measuremets that ca be obtaied by variatio of operatig frequecies ad icidece agles []. The o-fluctuatig compoet of errors represets a bias added to RCS measuremets. It appears i radar data due to calibratio errors ad time variatios of radar parameters []. Aother possible source of bias errors i soil parameter measuremets is the discrepacy betwee the predicted ad measured itesities of the scattered radar sigal. Errors of this type are observed for radar scatterig models that are ot accurate for the particular measuremet coditios ad soil characteristics, or for simplified empirical ad theoretical scatterig models. For example, the theoretical models of geometric optics, physical optics, small perturbatio method ad itegral equatio method, are oly valid for highly restricted surface types ad coditios [3] ad their applicatio for atural surfaces may produce bias errors. Cosiderably better agreemet with experimetal RCS measuremets is achieved for empirical scatterig models [7]. To establish the correspodece betwee soil properties ad specific RCS values we used the empirical model developed by Oh at al. [5], that estimates the ratio of specific RCS p=σ 0 HH/σ 0 VV, q=σ 0 HV/σ 0 VV at HH, VV ad HV radiowave polarisatios. The coefficiets p ad q of Oh at al. s model deped simultaeously o the rms height of surface roughess σ h ad soil dielectric costat ε [5] as follows: where Γ 0 θ i p = 1 π = 1 3Γ0 exp ( σ h [ 1 ( σ ] q = 0.3 Γ0 exp (1 ε ε + 1 h is the Fresel coefficiet, =π/λ is a waveumber, θ i is a icidece agle. Cosider the statistical characteristics of p ad q ratios for Oh at al. s empirical model. Assume that the followig coditios are satisfied: the radar is calibrated with high accuracy; the errors of the empirical scatterig model are small; the itesity of iterferece fluctuatios is much higher tha the itesity of equipmet iteral oise. I this case the bias compoet of RCS measuremet errors ad the equipmet oise have cosiderably lower itesity tha the iterferece fluctuatios of the scattered sigal ad their impact is ot sigificat. Suppose that idepedet measuremets I, =1, of scattered sigal itesity are available ad the estimates of the surface specific RCS are formed by averagig of I radar measuremets to suppress the iterferece

3 fluctuatios ad improve the accuracy. The, the RCS estimate is 1 I = i I i. ( The average value of the itesity I obeys a gamma distributio law with order parameter [8] ad has probability desity fuctio (pdf 1 p( I = Γ( σ I I / σ e, I 0 where Γ ( is the gamma fuctio, ad σ is a distributio parameter. The average itesity I is equal to <I>=σ ad its variace is var(i=σ /. The specific RCS ratios p ad q for the empirical model (1 are give by HH VV HV VV (3 p = I / I, q = I / I (4 where I HH, I VV ad I HV are sigal itesities at HH, VV ad HV polarizatios. To fid the statistical characteristics of p ad q ratio estimates we assume that the parameter σ of gamma distributio i (3 is equal to ad the average estimate I is scaled by ad I = I. The, radom variables I have a chi-squared distributio χ with = degrees of freedom. It ca be show that the ratio of two variables with chi-squared distributio χ obeys the F-distributio law [9] with the pdf give by Γ( pr ( x = Γ( / / x (1 + x. (5 F-distributio (4 has mea value m F = /(- ad 4 ( variace σ F =. Obviously, the statistics ( ( 4 p ad q i (4 also have F-distributios, but their mea ad variace values are differet. The mea values for p ad q are calculated as m p = pt, mq = qt (6 where p t ad q t are the true values of the itesity ratios. It follows from (6 that the estimates (4 are biased ad their magitudes deped o the umber of averaged measuremets. The variaces of p ad q statistics are ( ( σ p = p t, σ q = q t. (7 ( ( ( ( Accordig to (7 the variaces of p t ad q t ratio estimates deped o the umber of averaged measuremets ad the true values of estimated parameters. These results cofirm the fact that the iterferece oise is multiplicative. Whe the umber of averaged measuremets <3 the variace of sigal itesity ratio is ot defied, ad at least three idepedet measuremets are required to produce a reliable estimate. The pdfs for the F-distributio ad for the umber of measuremets =3 ad =00 are show i Figure p(x F-distributio Gaussia distributio x p(x (a F-distributio Gaussia distributio x (b Figure 1 : Pdfs of p ad q statistics for averaged measuremets: (a =3; (b =00 The Gaussia pdf with equal mea ad variace parameters is depicted i Figure 1 by a dashed lie. For =3 the form of the F-distributio is sigificatly differet from the Gaussia, but for =00 there is o sigificat differece betwee two distributios. For relatively large umbers of averaged samples the distributios of p ad q statistics are approximately Gaussia ad the estimates of soil parameters based o mea square error criterio are quasi-optimal, as for Gaussia distributio the least mea square estimate ad maximum lielihood estimate are idetical. 3 Assessmet of Data Fusio Errors For assessmet of soil parameter estimatio errors we have to mae assumptios about the algorithm used for multichael radar data fusio ad the model used to describe the radar sigal scatterig from the bare soil surface. We assume that a algorithm based o mea square error criterio [6] is used for radar data fusio ad there are o restrictios imposed o the choice of soil scatterig model. But it is expediet to choose a particular scatterig model for cosideratio. For predictig the radar sigal itesities we used Oh at al. s empirical model [5]. This provides a good match betwee the estimated ad true soil parameters for a wide rage of soil

4 types ad characteristics [6]. Oh at al. s model establishes the depedecies betwee p ad q RCS ratios at differet radiowave polarizatios ad the parameters of bare soil, such as rms height of surface roughess σ h ad soil dielectric costat ε. The dielectric properties of soil are fuctioally depedet o the volumetric moisture cotet m v, percetage of sad S ad clay C i soil ad other factors. To calculate the dielectric costat ε we used the watersoil mixture model preseted i [3], which establishes the depedece betwee the dielectric costat ε ad moisture cotet m v as ε(m v for the particular soil type ad compositio. The, the ratios of RCS at differet polarizatios are cosidered as fuctios of soil parameters p(σ h, m v ad q(σ h, m v. To assess the soil parameter measuremet accuracy we used a approach based o local liear approximatio of p ad q ratios as fuctios of soil roughess σ h ad moisture cotet m v parameters. We assume that the errors i p ad q measuremets are relatively small ad that ear to the true p t ad q t values the depedecies p(σ h, m v ad q(σ h, m v are liear: p( p( σ h, mv p( + ( σ h σ h p( + ( mv mv,0 mv q( q( σ h, mv q( + ( σ h σ h q(, mv,0 + ( mv mv,0 mv (8 where ad m v,0 are true soil parameter values. The aalytical expressios (8 are obtaied by Taylor expasio of p(σ h,ε(m v ad q(σ h,ε(m v equatios for Oh at al. s model, where the depedece ε(m v is give by the water-soil mixture model [3]. Soil parameters σ h ad m v are estimated by stadard least-squares method assumig that the errors of p ad q ratio measuremets are small ad the mea square error criterio ca be used for parameter estimatio [6]. Cosider the vector H = [ p 1,p,,pI, q1,q,,qj ] of radar measuremets for p ad q RCS ratios at differet operatig frequecies ad icidece agles. I geeral, the umber of measuremets for p ad q ratios ca be differet. The measuremet vector H is preseted as H = H + S = Z + δh + S (9 where Z is a vector of true values of p t ad q t ratios, δh is a vector of radom measuremet errors due to the sigal iterferece fluctuatios ad equipmet iteral oise. As it has bee show i sectio, the distributios of δh vector elemets are close to Gaussia. S is a vector of systematic errors, such as radar calibratio errors ad soil scatterig model errors. The elemets of vector S are split ito groups of similar errors, which correspod to differet radar sesors ad measuremet coditios. The systematic error vector is S=[s 1,s 1,,s,s,,s M,s M, ], where M is the umber of groups. Now, there is a eed to assess the impact of radom ad systematic errors o the accuracy of soil parameters estimatio, ad cosider the opportuity to detect ad compesate the bias error compoet. We have assumed i (8 that the fuctios p(σ h, m v ad q(σ h, m v, are liear close to the true parameters σ h ad m v. The vector of RCS ratios p t ad q t ca be expressed as Z=A N X, where X is a vector of soil parameters of legth (the vector X is equal to [σ h, m v,1] as we estimate the roughess ad moisture cotet parameters. A N is a array of dimesio N. The elemets of A N array are calculated as partial derivatives of Z=[z 1,z,,z N ] fuctios with respect to x 1,x,,x variables as z1 ( X0 / 1 z1 ( X0 /... Z( X0 A = = N X z N ( X0 / 1 z N ( X0 /... (10 where X 0 is a vector of true soil parameters. For the radar scatterig model i (1 ad water-soil mixture model i [3] the elemets of A N array i (10 are calculated by Ridder s umerical differetiatio method [10]. 3.1 Radom errors Cosider radar measuremets without systematic errors, whe S=0. The, applicatio of a least-squares method for estimatio of soil parameters gives the estimate of X vector as Xˆ N N N = ( A W A A W H (11 where W is a array of weightig coefficiets that weight the measuremets i proportio to their accuracy. The covariace matrix of least-squares estimate of Xˆ vector is the give by the expressio [11] R X ˆ = D R H D (1 where D = B A N W ad B = A N W A N. R H is the covariace matrix of radom measuremet errors δh for p ad q ratios. 3. Bias Errors The o-fluctuatig compoet of errors S 0 preseted i the vector of measuremets H has a impact o both the estimates of soil parameters Xˆ i (11 ad their statistical characteristics. Whe S 0 the mea of the estimated vector of parameters Xˆ is give by m X ˆ ( X + DS. (13 = It follows from (13 that o-fluctuatig error compoet adds a bias to soil parameter estimates. Sice the errors of soil parameter measuremets are evaluated as the differece betwee the estimated ad true parameters X

5 we ca assume that true parameters are equal to X =0. This assumptio simplifies the mathematical expressios for the assessmet of estimatio errors ad is valid for liear depedecies betwee measuremets H ad estimated parameters X. The, the measuremet errors ca be characterised by the matrix of secod momets. ˆ X Cosider the matrix of secod momets for vector Xˆ i (11. It ca be show that for the bias of magitude S i the measuremets the matrix of secod momets for the least-square estimates is give by X D R D + DSS ˆ = H Hece, the matrix ˆ X D ˆ X. (14 i (14 differs from the covariace matrix R Xˆ i (1 by the additioal compoet D SS D, which is proportioal to the magitudes of the systematic errors ad ca be foud from (13 as the square of bias D S. The matrix of secod momets also depeds o the covariace matrix ˆ X R H of radom measuremet errors. 3.3 Correctio of Bias Errors The accuracy of soil parameter measuremets ca be sigificatly improved by miimisig the ifluece of systematic errors. This is achieved by applicatio of efficiet radar calibratio techiques ad accurate radar scatterig models based o experimetal measuremets. But i may practical situatios the bias compoet of the residual errors may have a sigificat impact o the accuracy of soil parameter estimatio. I the presece of radar calibratio ad scatterig model errors the accuracy of radar data fusio algorithm ca be improved by estimatig the systematic error of vector S together with the uow soil parameters ad subsequet correctio of estimated errors. The elemets of vector S ca be cosidered as additioal estimated parameters ad added as ew elemets of vector Xˆ. The, the ifluece of bias errors is elimiated ad the secod compoet D SS D i equatio (14 becomes equal to zero. By addig ew estimated parameters we also icrease the impact of radom errors ad decrease the accuracy of estimates. The ifluece of radom errors is characterised by DR H D compoet i equatio (14. As the impact of radom errors is icreased, the improvemet of the accuracy of estimates must be questioed. To evaluate the improvemet achieved by estimatio of S error vector we used the matrix ˆ X of secod momets of least-squares estimates. The diagoal elemets of i (14 idicate ˆ X chages of soil parameter mea square errors whe the additioal parameters are estimated. Cosider the vector of estimated parameters X + exteded by a -elemet vector S of systematic errors. The least-squares method gives the estimate of the X + vector as Xˆ + = ( A + W A + A + W H. (15 N, N, N, The vector of estimates ca be preseted as X ˆ ~ ˆ + = [ xˆ 1, xˆ,..., xˆ, xˆ + 1, xˆ +,..., xˆ + ] = [ X X ], where ~ X is a vector of soil parameters ad X ˆ is a vector of systematic errors. The elemets of the exteded array A N,+ are foud as partial derivatives of fuctios Z = Z + S with respect to the exteded set of variables X + z1 z1 z1... Z ( X A N, + = = (16 X z N z N z N where X 0 is a vector of true values of parameters X +. It ca be show that the estimates X ˆ + i (15 obtaied by the least-squares method are ubiased [11]. The, the mea of X ˆ + is give by m( X ˆ + = X+ = [ X X ]. The covariace matrix for the least-squares estimate is give by ˆ = D X + R H D + + R (17 where D + = B+ A N + W ad B + = A N, + W A N, +. As metioed above, we ca assume that the true values of estimated parameters are equal to zero, ad X + =0. Taig ito accout that the estimates X ˆ + are ot biased, it ca be show that the covariace matrix R is equal to the matrix of secod momets. ˆ X + ˆ X + Hece, the matrix of secod momets is equal to = R ad ca be foud from equatio (17. Xˆ + Cosider X ˆ + ˆ X +, matrix as a bloc matrix ~ X +, Xˆ 1, composed of four matrix elemets. ˆ X, + 1 Xˆ, The bloc matrix elemet ~ of size is the X, covariace matrix for soil parameter measuremets X ~. Its aalytical expressio is obtaied by substitutio of the A array i equatio (17 ito a form of bloc array N, + [ A N, A N, ]. It ca be show that ~ matrix of X, secod momets is preseted as the sum of two compoets [1]

6 R + Q ~ = X ~, X, (18 where the compoet R ~ is equal to the covariace X, matrix R Xˆ i (1 for the least-squares estimates whe the regular compoet of errors is abset. The compoet Q i (18 idicates the deterioratio of the estimatio accuracy whe the umber of estimated parameters is exteded by the X ˆ vector of additioal parameters. Hece, the accuracy of soil parameter estimatio is restricted by the Q compoet i (18 whe the bias compoet of errors is estimated ad corrected, ad by the D SS D compoet i (14 whe the S vector is ot estimated simultaeously with soil parameters. To evaluate the efficacy of systematic error correctio we used the criterio based o compariso of the error secod momets ˆ ad X ~ give by the expressios (14 X, ad (17. After elimiatig the equal elemets R ~ ad X, R Xˆ i both expressios we eed to compare oly the D SS D ad Q compoets. For calculatio of the D SS D matrix we have to ow the true values of bias errors defied by vector S. But for the majority of practical applicatios the vector of bias errors S is ot ow a priori ad for evaluatio of the impact of systematic errors we use the approach based o approximatio. If the depedecies betwee the measured data ad estimated parameters are liear, for calculatio of D SS D matrix we ca use the estimates of systematic errors S ˆ = Xˆ give by equatio (15. Whe Oh at al. s radar scatterig model [5] is applied i combiatio with the water-soil mixture model [3] for soil parameter estimatio, the depedecies betwee the estimated parameters are oliear ad the equatio (15 based o liear least-squares method ca ot be applied. To estimate the vector of bias errors Ŝ for Oh at al. s scatterig model we use the oliear least-squares method based o umerical algorithms for objective fuctio miimisatio [7]. After Ŝ vector is foud the decisio is made about the importace of systematic error correctio by compariso of the D SS ˆ ˆ D ad Q matrices. The criterio we use to mae a decisio is tr( Q > tr( DSˆ Sˆ D (19 where tr( deotes trace of the matrix. If coditio (19 is satisfied we assume that the ifluece of S compoet i (9 is sigificat ad bias errors should be estimated as additioal parameters. Otherwise, oly the soil parameters are estimated. 4 Experimetal Results The proposed method establishes a relatioship betwee the soil parameter estimatio errors, radar measuremet errors ad radar parameters such as frequecy, polarisatio ad icidece agle. This relatioship ca be used to improve the accuracy of measuremets by correctio of radar calibratio errors ad optimisatio of radar parameters. The approach for mitigatio of systematic errors has bee preseted i Sectio 3.3. Now, we cosider how the proposed method ca be used to miimise soil parameter estimatio errors by proper selectio of radar operatig frequecies. For estimatig volumetric moisture cotet m v ad rms height of surface roughess σ h parameters we fuse multichael radar measuremets of a specific RCS tae at HH, VV ad HV polarizatios, two frequecies f 1 ad f selected i the rage 1-30GHz ad icidece agle θ=30. Assume that radar measuremets are tae for a uiform regio of bare soil with the followig characteristics: - the percetage of sad i soil S=0%; - the percetage of clay i soil C=9.7%; - paced soil desity ρ b =1311 G/m 3 ; - volumetric moisture cotet m v =0.3 g/cm 3 ; - rms height of surface roughess σ h = 0.4 cm. To establish the depedecies betwee the cosidered soil parameters, radar operatig parameters ad the itesities of soil scattered sigal we use Oh at al. s empirical scatterig model ad water-soil mixture model [3]. To estimate soil parameters we applied the oliear least-squares algorithm [6] that miimises the discrepacy betwee the measured ad Oh at al. s model predicted ratios p=σ 0 HH/σ 0 VV, q=σ 0 HV/σ 0 VV of specific RCS. The algorithm assumes weightig of p ad q radar measuremets accordig to their accuracy. The weightig coefficiets for W matrix (11 are selected i two differet ways: - the covariace array R H of radar measuremet errors δh of RCS ratios p ad q is ow a priori ad the weightig array is selected as W = R H ; - the accuracy of radar measuremets is ot ow i advace ad the measuremets are weighted equally as W=I, where I is a uit matrix. The measuremets of p ad q RCS ratios are cosidered to be statistically idepedet ad the covariace matrix R H to be diagoal. Statistical idepedece of p ad q measuremets is provided by usig differet radar operatig frequecies ad by varyig the radar positio ad operatig time itervals for the particular case whe the operatig frequecies f 1 ad f are idetical. The elemets of matrix R H are calculated from the expressio (7 for the variace of p ad q parameters. The experimetal results preseted i Figure ad Figure 3 illustrate the depedecies of secod momets for measuremet errors m (m v ad m (σ h for soil moisture cotet ad surface roughess parameters estimated by the proposed method. For calculatio of measuremet errors we used equatio (14 assumig that the compoets of S array are equal to zero whe the regular compoet of errors is abset. The radar operatig frequecies are selected i the iterval 1-30GHz ad the umber of averaged radar measuremets is equal to =00 for every frequecy. As it follows from expressio (7 for the umber of averaged data =00 the stadard deviatio of estimated RCS ratios is equal to σ p =0.1 p ad

7 m (σ h m (σ h f, GHz f, GHz (a (a m (m v m (m v f, GHz f, GHz (b (b m (σ h m (m v f, GHz (c (d f, GHz f 1, GHz Figure : secod momets of soil parameter estimatio errors for W = R H : (a soil roughess, S=0; (b soil moisture, S=0; (c soil roughess, S=0.0; (d soil moisture, S=0.0; Figure 3 : secod momets of soil parameter estimatio errors for W = I ad S=0: (a soil roughess; (b soil moisture σ q =0.1q for p ad q parameters respectively. Hece, the assumptio about liear depedecies of RCS with respect to soil moisture m v ad roughess σ h parameters is satisfied i the viciity of true p ad q ratios. For the depedecies i Figure the measuremets are weighted as W = R H ad the covariace matrix is ow a priori. Figures (a-(b show secod momets of measuremet errors for m v ad σ h soil parameter whe the bias error is abset. The depedecies are mootoic ad the accuracy of measuremets is improved whe both operatig frequecies f 1 ad f are decreased simultaeously. The miimum value of measuremet error for the cosidered iterval of operatig frequecies is achieved for f 1 =f =1. The secod momets of errors are equal to m (m v = ad m (σ h =0.001 for soil moisture ad roughess parameters respectively. Whe all the elemets of S array are equal to 0.01, the bias error of the same order as radom errors is added to all measuremets of p ad q ratios. The, the estimatio errors of m v ad σ h parameters are icreased for the whole rage of operatig frequecies. This is illustrated i Figures (c-(d. The form of the depedecies is ot chaged sigificatly. However, for the biased measuremets the fuctio m (σ h i Figure (c is icreased for small operatig frequecies ad reaches its miimum at frequecies of about 4 to 6 GHz. It follows from Figure 3 that the depedecies for the secod momets of errors m (m v ad m (σ h have more complex forms whe the measuremets are ot weighted

8 ad W=I. Whe the systematic errors are abset from radar measuremets the best estimatio accuracy of soil roughess parameter σ h is achieved for radar frequecies i the rage 10-15GHz ad for similar frequecies of lower magitude (see Figure 3(a. For the m v soil moisture parameter the best accuracy is provided i Figure 3(b for similar frequecies ad for the frequecies selected i the rage 0-30GHz. The bias error of magitude S=0.01 added to the measuremets of RCS ratios does ot chage the form of the fuctios i Figure 3 sigificatly. However, a more complicated depedecy betwee the accuracy of estimated parameters ad radar operatig frequecies is observed. The obtaied results agree with the experimetal data give i [5]. It is show i [5] that the sesitivity of p ad q ratios with respect to m v ad σ h soil parameters is decreased whe radar operatig frequecies are icreased. Hece, the accuracy of soil parameter measuremets ca be improved by applicatio of lower operatig frequecies. It also follows from the results preseted i Figure 3 that the estimatio accuracy ca be improved by selectig similar operatig frequecies if the covariace matrix is ot ow a priori ad measuremets are ot weighted. These recommedatios agree with the coclusios give i [] cocerig the ehacemet of specific RCS ratio sesitivity with respect to soil moisture parameter whe similar radar operatig frequecies are used for multichael measuremets at differet frequecies ad icidece agles. Hece, the efficacy of the proposed method for the assessmet of soil parameter estimatio errors is illustrated by the experimetal results for two-frequecy radar measuremets. 5 Coclusios A method based o local liear approximatio of specific RCS depedecies has bee proposed i this paper for assessmet of soil parameter estimatio errors, for fusio of multichael radar measuremets at differet operatig frequecies, polarisatios ad icidece agles. The proposed method does ot require ay specific assumptio about the type of radar scatterig model ad ca be used for radar data fusio based o both theoretical ad empirical models. The method evaluates the impact of radom ad systematic errors o the results of radar data fusio ad the accuracy of parameter estimatio. Its applicatio has bee cosidered for the assessmet of soil moisture ad roughess parameter estimatio errors for two-frequecy multipolarisatio radar measuremets. It has bee show that for Oh at al. s radar scatterig model the choice of radar operatig frequecies has a sigificat impact o radar data fusio ad parameter estimatio accuracy. I future wor our method will be used for remote sesig applicatios to assess the Earth surface parameter estimatio errors ad to optimise radar remote sesig system characteristics. Acowledgemets This research was supported by the Data Iformatio Fusio Defece Techology Cetre, Uited igdom, uder DTC Project 4.7. Refereces [1] M.I. Soli. Radar Hadboo. d ed., McGraw- Hill, New Yor, [] G.P. ulemi. Millimeter wave radar targets ad clutter. MA: Artech House, 003. [3] F.T. Ulaby, R.. Moore, A.. Fug. Microwave Remote Sesig: Active ad Passive, From Theory to Applicatios, vols. ad 3. Dedham, MA: Artech. House, [4] G.P. ulemi, N.G. Zerdev. Soil Erosio Effects i Microwave Scatterig from Bare Fields. Proceedigs of the 4-th Europea Microwave Coferece, vol. 1, pp , Caes, Frace, [5] Y. Oh,. Sarabadi, F.T. Ulaby. A empirical model ad a iversio techique for radar scatterig from bare soil surfaces. IEEE Tras. Geosciece ad Remote Sesig, vol. 30(, pp , March 199. [6] G.P. ulemi, A.A. Zelesy, A.A. urei, V.V. Lui. Solvig the Iverse Tas of Remote Sesig for Soil Parameter Estimatio. Proceedigs of the -d Russia Coferece Remote Sesig of Lad Covers ad Atmosphere by Aerospace Meas, vol. 1, pp.85-89, S-Petersburg, Russia, Jue 004. [7] G.P. ulemi. Multichael Remote Sesig Methods for Soil Parameters Estimatio. Proceedigs of the -d Russia Coferece Remote Sesig of Lad Covers ad Atmosphere by Aerospace Meas, vol. 1, pp.90-94, S-Petersburg, Russia, Jue 004. [8] C. Oliver, S. Quega. Uderstadig Sythetic Aperture Radar Images. Scitech Publishig Ic., 004. [9] E.W. Weisstei. F-Distributio. MathWorld--A Wolfram Web Resource. [10] W.H. Press, S.A. Teuolsy, W.T. Vetterlig, B.P. Flaery. Numerical Recipes i C++: The Art of Scietific Computig. d Ed., Cambridge Uiversity Press, 00. [11] N.N. Draper, H. Smith. Applied Regressio Aalysis. 3rd Ed., Joh Wiley & Sos, New Yor, [1] B.F. Zdaiu. Fudametals of Trajectory Measuremets ad Statistical Data Processig. Moscow, Sov. Radio, 1978 (i Russia.

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