remote sensing Article
Four-Stage Inversion Algorithm for Forest Height Estimation Using Repeat Pass Polarimetric SAR Interferometry Data Tayebe Managhebi, Yasser Maghsoudi *
ID
and Mohammad Javad Valadan Zoej
ID
Faculty of Geodesy and Geomatics, K. N. Toosi University of Technology, Tehran 19667-15433, Iran;
[email protected] (T.M.);
[email protected] (M.J.V.Z.) * Correspondence:
[email protected]; Tel.: +98-912-569-8068 Received: 29 May 2018; Accepted: 11 July 2018; Published: 25 July 2018
Abstract: This paper proposes a new method for forest height estimation using single-baseline single frequency polarimetric synthetic aperture radar interferometry (PolInSAR) data. The new algorithm estimates the forest height based on the random volume over the ground with a volume temporal decorrelation (RVoG+VTD) model. We approach the problem using a four-stage geometrical method without the need for any prior information. In order to decrease the number of unknown parameters in the RVoG+VTD model, the mean extinction coefficient is estimated in an independent procedure. In this respect, the suggested algorithm estimates the mean extinction coefficient as a function of a geometrical index based on the signal penetration in the volume layer. As a result, the proposed four-stage algorithm can be used for forest height estimation using the repeat pass PolInSAR data, affected by temporal decorrelation, without the need for any auxiliary data. The suggested algorithm was applied to the PolInSAR data of the European Space Agency (ESA), BioSAR 2007 campaign. For the performance analysis of the proposed approach, repeat pass experimental SAR (ESAR) L-band data, acquired over the Remningstorp test site in Southern Sweden, is employed. The experimental result shows that the four-stage method estimates the volume height with an average root mean square error (RMSE) of 2.47 m against LiDAR heights. It presents a significant improvement of forest height accuracy, i.e., 5.42 m, compared to the three-stage method result, which ignores the temporal decorrelation effect. Keywords: four-stage algorithm; forest height; mean extinction coefficient; polarimetric synthetic aperture radar interferometry; RVoG+VTD model; temporal decorrelation
1. Introduction Forest height is an important parameter in quantitative analysis of the carbon cycle at regional and global scales. Radar imagery provides us a great promise for the forest height estimation because of its global coverage and low weather sensitivity [1]. In recent years, the PolInSAR technique has been developed in terms of forest characterization based on the coherent combination of radar polarimetry and SAR interferometry technologies [2]. The capability of the PolInSAR technique results from its sensitivity to the 3-D distribution of scatterers in the vertical direction to separate the volumetric and surface scattering phase centers [3,4]. In the ideal case, the forest height can be estimated by scaling the phase difference between two interferograms corresponding to the volume-only and ground-dominant polarization states [3]. However, the phase centers of two selected polarization states depend on the propagation properties, such as wave extinction and the vertical structure of the canopy. In this regard, a physical model has been developed to invert the PolInSAR measurement into the forest height, taking into account the effective parameters of the wave and forest [5]. The random volume over
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ground (RVoG) model is a simple physical model, which considers both the volume layer and ground interactions. Accordingly, the vegetation layer is modeled as a homogeneous volume constituted by randomly-oriented particles by a constant wave attenuation in the media over a non-penetration ground [5]. The RVoG model addresses the complex coherence in each polarization state as a function of the volume thickness, the ground underlying phase, the mean extinction coefficient, and the surface-to-volume backscattering ratio. Thus far, different methods have been proposed to estimate the forest height based on the RVoG model. In this respect, a six-dimensional nonlinear optimization method was suggested using an iterative procedure [5]. According to the nonlinear nature of the problem, the most important disadvantage of this optimization method is that the results are often sensitive to the initial values of the variables. In order to overcome the above-mentioned disadvantages, a simple geometrical three-stage method was also proposed [4]. This method can estimate the ground phase, the forest height and the mean extinction coefficient without the need for a priori information and a time consuming iterative procedure. In this framework, this algorithm has been widely used in forest height estimation using the RVoG model due to its simplicity and time savings [6–9]. Basically, the RVoG model does not account for dynamic changes within the scene occurring in repeat pass PolInSAR data. For the purpose of the temporal decorrelation compensation, a two-layer model, random volume over ground with volume temporal decorrelation was applied [10]. The RVoG+VTD model incorporates the temporal decorrelation effect in the RVoG scattering model only in a very abstract way. Although the three-stage method can be applied for the inversion of the RVOG+VTD model, the accuracy of the inversion results depends on how well the extinction value is determined from an auxiliary data [10,11]. In reality, the RVoG+VTD model can be inverted using the three-stage method by fixing the mean extinction coefficient to a predefined value. In this framework, the accuracy of the estimated forest height depends on the particular value chosen for the mean extinction coefficient. In order to deal with this issue, a multi-baseline PolInSAR data was employed to invert the RVoG+VTD model without the need for an axillary data [12,13]. Since temporal decorrelation is one of the most common decorrelation resources over the vegetated land surface, a random motion over ground model (RMoG) has been developed to compensate for the temporal decorrelation in the coherence model. The RMoG model addresses the dynamic changes as a random motion of the scatterers with standard deviation that varies linearly along the vertical direction in the canopy [11,14]. In this framework, the RMoG model is characterized by two additional parameters, namely the canopy and ground motion standard deviation. In practice, the obtained solution depends on the initial values of the unknown parameters, strongly. Therefore, due to the nonlinear nature of the model, poor starting values lead to an inaccurate estimation [11]. In order to alleviate this problem, a supervised algorithm was proposed to invert the RMoG model to solve all unknown parameters [15]. It should be noted that, by tweaking the mean extinction coefficient value in the RVoG+VTD model, we are able to achieve an estimation performance similar to the ones of the RMoG result [11]. This research addresses the temporal decorrelation effect in the repeat pass PolInSAR data processing as a critical problem in the forest height estimation based on the RVoG+VTD model. As already noticed, in the absence of the predefined value for the volume temporal decorrelation or the mean extinction coefficient, the three-stage method is inefficient for the RVoG+VTD inversion. In this framework, we propose a new geometrical method for the RVoG+VTD retrieval using single-baseline PolInSAR data without the need for any auxiliary data, prior information or initial values of the variables. Hence, the advantage of the suggested method, which is termed four-stage algorithm, is the mean extinction coefficient estimation, independently. In other words, the proposed algorithm is used to overcome the limitation of the three-stage method in the RVoG+VTD inversion due to its capability of the biophysical parameters and temporal decorrelation coefficient estimation, geometrically. The structure of this paper is as follows: In Section 2, the selected PolInSAR dataset and the test site are introduced. Additionally, the RVoG model, three-stage inversion method, and the RVoG+VTD model are reviewed as the general underlying theory. Then, the proposed four-stage inversion algorithm is discussed. In Section 3, the results of the proposed method are analyzed by comparing
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comparing the results three-stage results against LiDAR heights. Finally, the research findings and their the three-stage against LiDAR heights. Finally, the research findings and their implications are implications discussed Section 4 and the limitations the proposed four-stage discussed in are Section 4 and in the advantages andadvantages limitationsand of the proposedoffour-stage algorithm are algorithm in areSection discussed discussed 5. in Section 5. 2. Materials and Methods In this section the selected PolInSAR dataset and the test site are introduced. Additionally, the underlying theory of the RVoG model, the conventional three-stage method, the the RVoG+VTD RVoG+VTD model, model, and the proposed four-stage method method are are discussed. discussed. 2.1. Test Site 2.1. Test Site and and PolInSAR PolInSAR Data Data Set Set Description Description A A pair pair of of fully fully polarimetric polarimetric L-band L-band airborne airborne data data from from the the BioSAR BioSAR 2007 2007 ESA ESA campaign campaign are are employed to validate the result of the suggested inversion algorithm. The utilized PolInSAR data have employed to validate the result of the suggested inversion algorithm. The utilized PolInSAR data ◦ 0 N, 13◦ 380 E) in Southern Sweden, as shown in been collected fromfrom the Remningstorp test site have been collected the Remningstorp test (58 site 28 (58°28′N, 13°38′E) in Southern Sweden, as shown Figure 1 [16]. in Figure 1 [16].
Figure 1. The location of the Remningstorp test site in Sweden [16].
About 1200 About 1200 ha ha in in the the test test site sitehave havebeen beencovered coveredby byproductive productiveforest forestinina afairly fairlyflat flatlandscape landscapeinina way that itsits height varies between 120120 andand 145145 m above the the mean sea sea level. The The selected L-band data a way that height varies between m above mean level. selected L-band pair have been acquired with 8 m spatial baseline and resolution of 0.74 m in range and 1.5 m in azimuth data pair have been acquired with 8 m spatial baseline and resolution of 0.74 m in range and 1.5 m direction. Two images Two haveimages been acquired on 31acquired March 2007 and 2 May2007 2007and with than one in azimuth direction. have been on 31 March 2 more May 2007 withmonth more temporal baseline. In the following, withmeasurements resolution of 0.5 m are used as reference than one month temporal baseline.LiDAR In the measurements following, LiDAR with resolution of 0.5 m height to as evaluate theheight results.toAll processing has been to ahas partbeen of the imagetoconsists ofthe 15 are used reference evaluate the results. Allapplied processing applied a part of particular stands. Theparticular delineation of theThe used forest stands hasused beenforest presented Figure image consists of 15 stands. delineation of the standsinhas been2.presented in Figure 2.
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Figure2.2. 2.Images (a)E-SAR E-SARimage imageininthe thePauli Paulibasis, basis,(b) (b)selected selectedarea area(the Figure Imagesofofthe thestudy studyarea. area.(a) (theboundary boundary Figure the stands has been represented by red closed curves). ofthe thestands standshas hasbeen beenrepresented representedby byred redclosed closedcurves). curves). ofof
2.2.Random RandomVolume Volumeover overGround GroundModel Model 2.2. 2.2. Ground Model simplephysical physicalmodel interpretthe theforest foreststructure structure Random Random volume volumeover over ground ground isis aa simple simple physical model toto interpret interpret the forest structure information by making use of the complex coherence in different polarization states [5,17]. Figure3,3, information by making use of the complex coherence in different polarization states [5,17]. Figure schematic representation physical interpretation the RVoG model. Asshown shown Figure the isisaaschematic ofofof physical interpretation ofofthe RVoG model. As ininFigure 3,3,the schematicrepresentation representation physical interpretation of the RVoG model. As shown in Figure 3, RVoG model uses an exponential structure function as [1]: RVoG model usesuses an exponential structure function as [1]: the RVoG model an exponential structure function as [1]: 22σz σ2zσz cos cos cos e θθ00θ 0,
==ee fff((z( )z) )=
(1) (1) (1)
,,
whereinσ the constant mean extinction coefficient and isincidence incidence angle. wherein the constant mean extinction coefficient and angle. wherein σσisisis the constant mean extinction coefficient and θ 0θθ0is0isincidence angle.
Figure 3.Two-layer Two-layer physicalmodel modeland andits itsstructure structurefunction functionrepresentation representation[18]. [18]. Figure Figure3.3. Two-layerphysical physical model and its structure function representation [18].
Accordingtotothe theno noground grounddecorrelation decorrelationassumption, assumption,the thecomplex complexcoherence coherenceininaaspecific specific According According to the no ground decorrelation assumption, the complex coherence in a specific polarization state is expressed as [5]: polarization state is expressed as [5]: polarization state is expressed as [5]:
( w) ) γγ ++μμ( w ( w) )==eeiϕiϕiϕ γvvv + µ(w) , , γγ( w γ(w) = e 0 11++μμ( w ( w) ) , 1 + µ(w) 0 0
(2) (2) (2) iϕ
) represents inwhich which γ (w representsthe the interferometriccoherence coherence in a specificpolarization polarization state,eeiϕiϕ0 00 isisthe the inin whichγγ(w (w)) represents theinterferometric interferometric coherenceinina aspecific specific polarizationstate, state, e is the γvvvrefers refersto tothe thecomplex complexvolume volumecoherence, coherence,and andμµμ((w theeffective effective groundcomplex complexcoherence, coherence,γγ (ww)))isisisthe ground ground complex coherence, refers to the complex volume coherence, and the effective ground-to-volumebackscattering backscatteringratio ratiowhich whichisis the the only only polarization dependent parameter only polarization polarization dependent dependent parameter parameter inin in ground-to-volume Equation (2). By isolating the polarization dependent parameter in a single term, Equation (2) can be Equation (2). Equation Equation (2). By isolating the polarization dependent parameter in a single term, Equation (2) can be rewritten as [5]: rewrittenas as[5]: [5]: rewritten µ(w) γ(w) = eiϕ0 (γv + L(w)(1 − γv )) , L(w) = μ ( w) (3) μ ( w) ϕ i ( w) )==eeiϕ0 (0 γ(γv v++LL( w ( w)()(11−−γγv )v )) ), ,LL( w ( w) )== 1 + µ(w) (3) γγ( w (3) ( w) ) 11++μμ( w
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where L(w) lies in the range 0 ≤ L(w) ≤ 1, with limits occurring at one end for pure volume scattering (µ(wv ) = 0) and at the other for pure surface polarization state (µ(ws ) = ∞). According to Equation (3), the coherence values in different polarization states lie along a straight line. The volume-only coherence is also given explicitly as a function of the constant mean extinction coefficient and the forest height, hv as follows [5]: Z hv 2σz 2σ γv = eikz z e cos θ0 dz (4) 2σhv cos θ0 (e cos θ0 − 1) 0 where k z represents the vertical wave number which can characterized as [1]: kz =
4πB⊥ λR sin θ0
(5)
wherein B⊥ is the perpendicular spatial baseline, λ is the signal wave length, and R is the slant range. Equations (3) and (4) show a nonlinear relation between four unknown parameters (i.e., hv , σ, ϕ0 and µ(w)); in the RVoG model. As discussed in Section 1, several methods have been developed for forest height inversion based on the RVoG model, such as DEM differencing [3], the six-dimensional nonlinear optimization method [5], and the three-stage method [4]. The three-stage algorithm has been widely used in forest height estimation due to its simplicity and time saving [6–8]. In this framework, we used the three-stage method as our basis algorithm. 2.3. Three-Stage Inversion Method In the last two decades several model based forest height inversion methods have been suggested using the single-baseline PolInSAR configuration [3,4,11,19]. Among these methods the three-stage algorithm is a geometrical straightforward approach without the need for auxiliary data. In this respect, the three-stage algorithm has been one of the most frequently used method in single or dual baseline inversion method [6–8,20]. The above-mentioned method breaks the inversion procedure down into three separate steps, least squares line fit, ground phase estimation and, finally, the canopy height and the mean extinction coefficient estimation. Based on the RVoG model, the coherence signature in a specific resolution cell, which contains only two different phase centers, is a straight line in the complex unit circle [21]. In this respect, the first step of the three-stage method is the coherence line fitting using the complex coherence value in Pauli basis. Thereafter, the fitted line can be used to secure the intersection points between the coherence line and the complex unit circle (CUC) as the ground coherence candidates. In other words, according to the no ground decorrelation assumption in the RVoG model, two intersection points between CUC and the coherence line are the ground coherence candidates with λ g = 1, as shown in Figure 4. The main purpose of the second step is to choose the ground coherence between two candidates. Based upon the scattering physics, a strong scattering contribution from the ground is present in the direct or dihedral ground backscattering, while the canopy scattering has a strong HV signal [4,22]. In this regards, it is very unlikely that the weakest ground to volume scattering ratio will arise in the HV channel. Hence, it is reasonable to expect that the ground coherence will be ranked furthest away in distance from the HV coherence along the coherence line as shown in Figure 5. Finally, in the third step, two remained unknown parameters (i.e., hv and σ); are estimated according to the Equation (4). For the purpose of using Equation (4), an observed polarization state is employed as the volume only channel. In doing so, the three-stage algorithm selects the observed volume only coherence among the common polarization states, such as HH, VV, HH-VV, HH+VV, and HV. Although Equation (4) provides two separate equations based on the observed volume only coherence, the non-linear nature of the problem leads to the need for the initial values of the hv and σ. In order to alleviate this problem, the three-stage method proposed a geometrical solution to estimate the correspondence volume height and the mean extinction coefficient. Figure 4 depicts the volume coherence loci for two constant mean extinction
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coherence, the non-linear nature of the problem leads to the need for the initial values of the hvhand coherence, the non-linear nature of the problem leads to the need for the initial values of the v and σσ. In order to alleviate this problem, the three-stage method proposed a geometrical solution to . In order to alleviate this problem, the three-stage method proposed a geometrical solution to estimate the correspondence volume height and the mean extinction coefficient. Figure 4 depicts the estimate correspondence volume height and the mean extinction coefficient. Figure 4 depicts Remote Sens.the 2018, 10, 1174 6 ofthe 16 volume coherence loci for two constant mean extinction coefficient values using Equation (4). The volume coherence loci for two constant mean extinction coefficient values using Equation (4). The coherence curves and the coherence line intersection points are the calculated volume only coherence coherence curves and the coherence line intersection points are the calculated volume only coherence coefficientwhich valuesare using Equation (4). The coherence and(LUT), the coherence line intersection points σ candidates employed to pre-calculate a lookcurves up table as a function of the h and candidates which are employed to pre-calculate a look up table (LUT), as a function of the vh and σ are the calculated volume only coherence candidates which are employed to pre-calculate av look up . It is worth stressing that, the three-stage method assumes γ HV is the observed volume only γ HVthe . It is(LUT), worthasstressing that, thehvthree-stage assumesthat, isthree-stage the observed volume only table a function of the and σ. It is method worth stressing method assumes coherence which has the upper phase center among the common polarization states. Since the γ volume only coherence which has the phase center among the Since common coherence which has the upper phase center among the upper common polarization states. the HV is the observed , σ = σ ) generates a negative in the observed intersection point corresponding to γ vγ ( h μ polarization states. Since the intersection point corresponding to γ ( h , σ = σ ) generates a negative 1 ) generatesv a vnegative 1 μ in the observed intersection point corresponding to v (vh v , σ = σ 1 µ in thecoherence, observed volume coherence, it cannot be a physical solution.allInintersection this regard, points all intersection volume it cannot be a physical solution. In this regard, on the volume coherence, it cannot be a physical solution. In this regard, all intersection points on the points on the line segment are used tothe pre-calculate the the LUT. Finally, the ambiguous lineambiguous segment are used to pre-calculate LUT. Finally, estimation of estimation the volumeof ambiguous line segment are used to pre-calculate the LUT. Finally, the estimation of the volume the volume height and the meancoefficient extinctioncan coefficient can be by comparing the calculated height and the mean extinction be secured bysecured comparing the calculated volume height and the mean extinction coefficient can be secured by comparing the calculated volume volume coherences and the observed volume coherence. In this the respect, the volume height the coherences and the observed volume coherence. In this respect, volume height and the and mean coherences and the observed volume coherence. In this respect, the volume height and the mean mean extinction coefficient, which minimize the distance between the calculated coherence and the extinction coefficient, which minimize the distance between the calculated coherence and the extinction coefficient, which minimize the distance between the calculated coherence and the observedvolume volumecoherence coherenceare arethe theinversion-solution inversion-solutioncouple. couple. observed observed volume coherence are the inversion-solution couple.
Figure 4. The three-stage algorithm procedure representation. Blue, red, and green circles depict the Figure Figure 4. 4. The The three-stage three-stage algorithm algorithm procedure procedure representation. representation. Blue, Blue, red, red, and and green green circles circles depict depict the the coherence values in the Pauli basis. The black line is the coherence line and the ground coherence coherence coherence values values in in the the Pauli Pauli basis. basis. The The black black line line is is the the coherence coherence line line and and the the ground ground coherence coherence iϕ ϕ. The andeiϕ ambiguous line segment been shown candidates e 1iϕ. 1iThe candidateshave havebeen beenshown shownby byeeiϕ00iϕand ambiguous line segment hashas been shown by by redred line. candidates have been shown by e 0 and e 1 . The ambiguous line segment has been shown by red TheThe black stars are are the the fixed extinction coherence lociloci [7]. [7]. line. black stars fixed extinction coherence line. The black stars are the fixed extinction coherence loci [7].
Figure 5. Schematic representation of the relative location of the coherence values in the Pauli states Figure 5. 5. Schematic Schematic representation representation of of the the relative relative location location of of the the coherence coherence values values in in the the Pauli Pauli states states Figure along the coherence line [7]. along the coherence line [7]. along the coherence line [7].
2.4. Random Volume over Ground with Volume Temporal Decorrelation 2.4. Random Random Volume Volume over over Ground Ground with with Volume Volume Temporal TemporalDecorrelation Decorrelation 2.4. The temporal decorrelation is a major source of decorrelation in the repeat pass interferometry The temporal temporal decorrelation decorrelation is is aa major major source source of of decorrelation decorrelation in in the the repeat repeat pass pass interferometry interferometry which The limits the interferometric capabilities of data. In general, the dynamic changes caused by wind which limits the capabilities ofof data. In general, the dynamic changes caused by wind which limits theinterferometric interferometric capabilities data. In general, dynamic changes and other changes such as vegetation growth, seasonal variation and the manmade changes cancaused severelyby and other changes such as vegetation growth, seasonal variation and manmade changes can severely windthe andcoherence other changes such as vegetation growth,the seasonal and manmade changes can affect measurements. Consequently, repeatvariation pass PolInSAR retrieval requires affect the coherence measurements. Consequently, the repeat pass PolInSAR retrieval requires severely affect the coherence measurements. the repeat PolInSAR requires modelling the temporal decorrelation effect.Consequently, Since the RVoG modelpass does not takeretrieval the temporal modelling the the temporal temporal decorrelation decorrelation effect. effect. Since the RVoG model does does not not take take the the temporal temporal modelling Since the RVoG model decorrelation effect into account, the estimated volume height is fairly overestimated due to the decorrelation effect effect into into account, account, the the estimated estimated volume volume height height is is fairly overestimated overestimated due due to to the the decorrelation temporal decorrelation effect. To cope with this limitation, two fairly complementary models were temporaldecorrelation decorrelationeffect. effect. To cope withlimitation, this limitation, two complementary models were temporal cope with[11,14] this complementary models were suggested, suggested, i.e., RVoG+VTD [10]To and RMoG models. two As discussed in Section 1, Papathanassiou suggested, i.e., RVoG+VTD [10] and RMoG [11,14] models. As discussed in Section 1, Papathanassiou i.e.,Cloude RVoG+VTD and RMoG [11,14] models. As discussed Papathanassiou and Cloude and have[10] suggested a supervised three-stage methodintoSection invert 1,the RVoG+VTD model [10] and Cloude have suggested athree-stage supervisedmethod three-stage method to invert the model RVoG+VTD model [10] have suggested a supervised to invert the RVoG+VTD [10] and Lavalle and Lavalle and Hensley have proposed an eight dimensional nonlinear optimization method for and Hensley Lavalle and have proposed an eightnonlinear dimensional nonlinearmethod optimization method for and haveHensley proposed an eight dimensional optimization for RMoG model inversion [11]. In this paper, we addressed the temporal decorrelation based on the RVoG+VTD model.
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To compensate the temporal decorrelation effect in the RVoG model, two real multiplying factors were added to the RVoG model as [10]: td iϕ0 | γtv | γv + γtg µ ( w ) (6) γ(w) = e 1 + µ(w) where |γtv | and γtg are the temporal decorrelation coefficients of volume and ground coherences, respectively, both ranging from zero to one and γ(w)td is the complex coherence in a desired polarization state taking the temporal decorrelation into account. Generally, the most common temporal decorrelation effect comes from the wind-induced movement of the canopy layer within a few-day temporal baseline [23]. With such an argument, the scattering properties of the ground is assumed constant during the time, i.e., γtg = 1. In this respect, Equation (6) can be simplified with only the volume temporal decorrelation multiplying factor (i.e., RVoG model with volume temporal decorrelation, RVoG+VTD) as [10]: γ(w)td = eiϕ0
|γtv |γv + µ(w) 1 + µ(w)
(7)
where γv is characterized as Equation (4). Furthermore, Equation (7) can be rewritten as: γ(w)td = eiϕ0 (|γtv |γv + L(w)(1 − |γtv |γv )) , L(w) =
µ(w) 1 + µ(w)
(8)
Based upon Equation (8), the |γtv | leads to a degradation of the amplitude of the volume coherence and it does not affect the interferometric phase center [10,15]. Additionally, Equation (8) shows that the coherence loci in the complex plane is a straight line. The coherence lines correspond to the RVoG and the RVoG+VTD models are depicted in Figure 6 as the blue line and dotted blue line respectively. As shown in Figure 6, the temporal decorrelation reduces the coherence amplitude in all polarization states. In this framework, the volume-only coherence point moves radially towards the origin because of the temporal decorrelation effect. In other words, by varying |γtv | the coherence line is rotated about the ground coherence point. In the extreme case, |γtv | = 0, the volume coherence fall into the origin of the CUC and in the other extreme case of no temporal decorrelation the RVoG+VTD model conforms to the RVoG model. In order to invert the RVoG+VTD model, the ground phase can be estimated by implementing the first two stages of the conventional three-stage method. However, the amplitude of the observed volume coherence corresponds to the product of the amplitude of the pure volume only coherence, |γv |, and |γtv |. Hence, the observed volume coherence should be shifted radially towards the top to compensate the temporal decorrelation effect. In other words, as there is no knowledge about the volume temporal decorrelation value, |γtv |, the radius segment beyond |γtv |γv up to the complex unit circle is the corresponding ambiguous line segment of the volume-only coherence. Accordingly, the forest height estimation, based on the RVoG+VTD model using single-baseline PolInSAR data, is insoluble without the information about the mean extinction or volume temporal decorrelation value. In order to overcome this challenge, employing a fixed mean extinction coefficient is suggested [10]. As shown in Figure 7, the volume-only coherence point can be found as the intersection point between the fixed extinction volume coherence curve and the constant volume coherence phase line, i.e., ϕ = ϕ HV . Finally, the forest height and the volume temporal decorrelation corresponding to the calculated volume only coherence are the inversion-solution couple. In practice, the accuracy of the estimated height depends on the selected constant mean extinction value. In other words, the results are changeable by tweaking the mean extinction value. This makes the RVoG+VTD inversion with a single-baseline PolInSAR configuration a challenge [11]. Alternatively, the RVoG+VTD can be inverted using a dual baseline inversion algorithm [12].
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are are changeable changeable by by tweaking tweaking the the mean mean extinction extinction value. value. This This makes makes the the RVoG+VTD RVoG+VTD inversion inversion with with aa single-baseline PolInSAR configuration be single-baseline configuration aa challenge challenge [11]. [11]. Alternatively, Alternatively, the the RVoG+VTD RVoG+VTD can can Remote Sens. 2018, 10,PolInSAR 1174 8 ofbe 16 inverted using a dual baseline inversion algorithm [12]. inverted using a dual baseline inversion algorithm [12].
Figure Geometrical interpretation of the multiplying Figure Figure 6. 6. Geometrical Geometrical interpretation interpretation of of the the effect effect of of the the volume volume temporal temporal decorrelation decorrelation multiplying multiplying factor. The coherence loci in the RVoG shown solid blue and factor. The coherence loci in the RVoG and RVoG+VTD models have been shown in solid factor. The coherence loci in the RVoG and RVoG+VTD models have been shown in solid blue blue and and dotted blue lines. The dotted red lines show the ambiguous radial segment for Pauli basis coherences. dotted dotted blue blue lines. lines. The The dotted dotted red red lines lines show show the the ambiguous ambiguous radial radial segment segment for for Pauli Pauli basis basis coherences. coherences.
Figure representation [10]. Figure7. Geometrical representationof theRVoG+VTD RVoG+VTDinversion inversionusing usingσ σ000[10]. σσ === σσ Figure 7.7.Geometrical Geometrical representation ofofthe the RVoG+VTD inversion using [10].
Four-Stage Inversion Inversion Method 2.5. 2.5. Four-Stage Four-Stage Inversion Method Method As is is not able to estimate the the biophysical parameters and noticed, the three-stage method able to biophysical parameters As already alreadynoticed, noticed,the thethree-stage three-stagemethod method is not not able to estimate estimate the biophysical parameters the temporal decorrelation multiplying factor in the RVoG+VTD model. Inmodel. order to and the temporal decorrelation multiplying factor in In order andvolume the volume volume temporal decorrelation multiplying factor in the the RVoG+VTD RVoG+VTD model. Inovercome order to to this issue, Papathanassiou and Cloude proposed to use a fixed mean extinction coefficient and then overcome this issue, Papathanassiou and Cloude proposed to use a fixed mean extinction coefficient overcome this issue, Papathanassiou and Cloude proposed to use a fixed mean extinction coefficient to estimate two remaining variables variables (i.e., the forest height and the real valued volume and then estimate the (i.e., forest height and the valued volume and then to tothe estimate the two two remaining remaining variables (i.e., the the forest height and the real real valuedtemporal volume decorrelation); geometrically [10]. Intuitively, the accuracy of the results depends on how good the temporal decorrelation); geometrically [10]. Intuitively, the accuracy of the results depends on temporal decorrelation); geometrically [10]. Intuitively, the accuracy of the results depends on how how guess is for the mean extinction coefficient value. In order to alleviate this problem, we propose a good good the the guess guess is is for for the the mean mean extinction extinction coefficient coefficient value. value. In In order order to to alleviate alleviate this this problem, problem, we we four-stage algorithm algorithm which estimates reliable mean extinction coefficientcoefficient in the third and propose which estimates the reliable mean in the propose aa four-stage four-stage algorithm which the estimates the reliable mean extinction extinction coefficient instage the third third finally estimates the forest height and the real valued volume temporal decorrelation multiplying stage and finally estimates the forest height and the real valued volume temporal decorrelation stage and finally estimates the forest height and the real valued volume temporal decorrelation factor in the factor last stage. Inlast other words, the four-stage algorithm extends the three-stage procedure to multiplying in stage. In words, four-stage algorithm extends three-stage multiplying factor in the the last stage. In other other words, the the four-stage algorithm extends the the three-stage invert the RVoG+VTD As shown in Figure 6, the temporal decorrelation, doesn’t affect procedure to RVoG+VTD model. As in Figure the temporal procedure to invert invert the themodel. RVoG+VTD model. As shown shown involume Figure 6, 6, the volume volume temporal decorrelation, decorrelation, the coherence locus. Therefore, first two steps of the proposed method is carried out similar to out the doesn’t affect the coherence locus. Therefore, first two steps of the proposed method is doesn’t affect the coherence locus. Therefore, first two steps of the proposed method is carried carried out conventional algorithm in order to estimate theto underlying phase. similar conventional three-stage algorithm in estimate the underlying similar to to the the three-stage conventional three-stage algorithm in order order toground estimate the ground ground underlying phase. phase. the mean extinction coefficient is a function of the forest biophysical properties such as In general, mean extinction coefficient is a function of the forest biophysical properties such In general, the mean extinction coefficient is a function of the forest biophysical properties such volume height and forest density and the sensor parameters such as wave frequency. The higher mean as as volume volume height height and and forest forest density density and and the the sensor sensor parameters parameters such such as as wave wave frequency. frequency. The The higher higher extinction, which is took place at higher frequencies, leads to an effective backscattering from the top mean extinction, which is took place at higher frequencies, leads to an effective backscattering mean extinction, which is took place at higher frequencies, leads to an effective backscattering from from of the canopy [24]. Based on this, the mean extinction coefficient value and the penetration depth are the top of the canopy [24]. Based on this, the mean extinction coefficient value and the penetration the top of the canopy [24]. Based on this, the mean extinction coefficient value and the penetration inversely related. Also, the volume decorrelation increases with the penetration depth. Therefore, the volume coherence amplitude and the penetration depth are indirectly related. Figure 8 represents
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Remote Sens. 10, 1174related. Also, the volume decorrelation increases with the penetration depth. 9 of 16 depth are 2018, inversely Therefore, the volume coherence amplitude and the penetration depth are indirectly related. Figure 8therepresents the complex volume coherence the CUC for varying extinction complex volume coherence variation insidevariation the CUC inside for varying extinction coefficients when
=◦ .0.1567 θ 0 = 45°. A look at Figure 8 reveals that the loci fall in coefficients when kz = 0.1567 and θ0 k =z45 A lookand at Figure 8 reveals that the coherence loci fall incoherence to the trigonometric circle owing to the extinction coefficient reduction. On the other hand,On thethe volume to the origin trigonometric circle origin owing to the extinction coefficient reduction. other scattering hand, the phase center moves towards the top oftowards the canopy according to the higher mean extinction volume scattering phase center moves the top of the canopy according to the higher value, mean which corresponds to the low penetration depth. Consequently, the mean extinction coefficient and extinction value, which corresponds to the low penetration depth. Consequently, the mean extinction the complex volume coherence are significantly related to each other. coefficient and the complex volume coherence are significantly related to each other.
Figure Figure 8. 8. The The volume volume coherence coherence loci loci representation representation in in the the complex complex unit unit circle circle [7]. [7].
With such an argument, Managhebi et al. proposed to employ the relative location of the With such an argument, Managhebi et al. proposed to employ the relative location of the observed observed volume coherence on the coherence line to limit the range of the mean extinction coefficient volume coherence on the coherence line to limit the range of the mean extinction coefficient in an in an improved three-stage method [7]. In this framework, a geometrical index was suggested to improved three-stage method [7]. In this framework, a geometrical index was suggested to interpret interpret the relative location of the observed volume coherence, γ HV , on the coherence line as [7]: the relative location of the observed volume coherence, γHV , on the coherence line as [7]:
D.I = = D.I
A.L A.L VV.L .L
(9) (9)
wherein thedistance distanceratio ratioindex, index, and arethe the ambiguous length visible A.Land D.I isisthe V.Lare wherein D.I A.L V.L ambiguous lineline length andand the the visible line line length respectively, as shown in Figure 9. In this geometrical index, the has been normalized A. L length respectively, as shown in Figure 9. In this geometrical index, the A.L has been normalized by the by theThe index. The geometrical is a real non-negative value occurring with limits V.L to generalize V.Lthe to generalize the index. geometrical index is aindex real non-negative value with limits at occurring at one end for the case of no penetration ( ) and at the other by pure surface scattering D . I = 0 one end for the case of no penetration (D.I = 0) and at the other by pure surface scattering (D.I = ∞) (when the penetration depthtoisthe equal to theheight volume height theoretically. depicts D.I = the ∞ ) when penetration depth is equal volume theoretically. Figure 10 Figure depicts10 the mean the mean extinction value by calculated by the method three-stage 75 pixels of three simulated extinction value calculated the three-stage in 75method pixels ofinthree simulated PolInSAR data PolInSAR data for three species (i.e., deciduous and two types of pine); versus D.I. As ( ) h = 20 m species (i.e., deciduous and two types of pine); versus D.I. As expected, the mean (hv = 20 m) for three v extinction the value andextinction D.I value are inversely As inversely shown in Figure geometrical index expected, mean value and D.Irelated. value are related.10, Asthe shown in Figure 10, can the be further utilized extinction coefficient estimation effectively. Therefore, in this research, we geometrical index for canmean be further utilized for mean extinction coefficient estimation effectively. defined theinmean extinction coefficient as mean the following linear function the distancelinear ratio index as: Therefore, this research, we defined the extinction coefficient asof the following function of the distance ratio index as: σ = aD.I + b (10) (10) σ = aD.I + b where σ σ isis the coefficient, D. D.II is distance ratio ratio index, index, aa and and bb are model where the mean mean extinction extinction coefficient, is the the distance are the the model parameters. To To compute compute the the variables variables of of the the linear linear model modelwe weused usedaareal realL-band L-bandPolInSAR PolInSARdata datapair. pair. parameters. Figure 11 represents the flowchart for estimating the unknown parameters of Equation (10) using the Figure 11 represents the flowchart for estimating the unknown parameters of Equation (10) using the least squares method. least squares method.
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Figure 9. The ambiguous line line segment and the line segment on theon coherence line [7].line [7]. Figure 9. The ambiguous segment andvisible the visible line segment the coherence Figure 9. The ambiguous line segment and the visible line segment on the coherence line [7]. Figure 9. The ambiguous line segment and the visible line segment on the coherence line [7].
Figure 10. The mean extinction coefficient versus distance ratio index scatterplot. The scatterplot has Figure mean extinction coefficientversus versus distanceratio ratio index index scatterplot. scatterplot. The scatterplot has Figure 10. 10. TheThe mean extinction coefficient The scatterplot been generated using three different forest speciesdistance (i.e., deciduous and two types of pine); in threehas been three different forest species (i.e., deciduous and two of pine); Figure 10.generated The mean using extinction coefficient versus distance ratio index scatterplot. The types scatterplot has in three been generated usingdata three simulated PolInSAR [7].different forest species (i.e., deciduous and two types of pine); in three beensimulated generatedPolInSAR using three different data [7]. forest species (i.e., deciduous and two types of pine); in three simulated PolInSAR data [7]. simulated PolInSAR data [7].
Figure Flowchart for finding the unknown parameters the linear Figure 11. 11. Flowchart for finding the unknown parameters of the of linear model.model.
Figure Flowchart findingthe theunknown unknownparameters parameters of the linear Figure 11.11. Flowchart forforfinding linear model. model.
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summary, following step-by-step describes linear model parameters, InIn summary, the the following step-by-step outlineoutline describes finding finding the linearthe model parameters, a and b.
a and b .
1.
1.
2. 2. 3. 3. 4.4. 5.5. 6.6. 7.7.
Fit the least square line on the Pauli basis coherences and find the intersection points between the Fit the least square line on the Pauli basis coherences and find the intersection points between fitted line and the CUC as the ground coherence candidates. the fitted line and the CUC as the ground coherence candidates. Choose the ground underlying phase between two candidates. Choose the ground underlying phase between two candidates. Extract the forest height from LiDAR measurement as the reference height. Extract the forest height from LiDAR measurement as the reference height. Calculate Calculatethe theD.I D.Ifor forthe theselected selectedpixels pixelsusing usingEquation Equation(9). (9). Calculate the fixed height coherence locus for all (4). Calculate the fixed height coherence locus for allselected selectedpixels pixelsusing usingEquation Equation (4). Estimate the mean extinction coefficient corresponding to the intersection point Estimate the mean extinction coefficient corresponding to the intersection pointbetween betweenthe the volume coherence locus and the coherence line for all selected pixels. volume coherence locus and the coherence line for all selected pixels. Calculate Calculatethe themodel modelparameters parametersusing usingthe theleast leastsquare squaremethod methodbased basedononEquation Equation(10). (10).
Afterwards, Equation (10) employed the third stage the four-stage algorithm compute Afterwards, Equation (10) is is employed inin the third stage ofof the four-stage algorithm toto compute the mean extinction coefficient in pixel. each Figure pixel. Figure 12the shows the flowchart of thealgorithm four-stage the mean extinction coefficient in each 12 shows flowchart of the four-stage algorithm for forest heightbased estimation on the model. RVoG+VTD model. for forest height estimation on thebased RVoG+VTD
Figure 12.12. Flowchart ofof the proposed four-stage inversion algorithm. Figure Flowchart the proposed four-stage inversion algorithm.
illustratedininFigure Figure12, 12,the the first two steps the proposed four-stage inversionmethod methodare are AsAsillustrated first two steps ofof the proposed four-stage inversion carried out similar to the conventional three-stage algorithm to estimate the ground underlying carried out similar to the conventional three-stage algorithm to estimate the ground underlying phase. phase. main between difference proposed and the conventional three-stage The main The difference the between proposed the method and themethod conventional three-stage algorithm lies in algorithm lies in the mean extinction coefficient estimation. As discussed in Section 2.4, the threethe mean extinction coefficient estimation. As discussed in Section 2.4, the three-stage algorithm uses stage algorithm uses a fixed mean extinction coefficient to reduce the number of the variables in the a fixed mean extinction coefficient to reduce the number of the variables in the RVoG+VTD model. RVoG+VTD model. This ismethod while the four-stage calculatescoefficient the mean based extinction coefficient This is while the four-stage calculates the method mean extinction on the signal based on the signal penetration and the relative location of the volume coherence on the coherence penetration and the relative location of the volume coherence on the coherence line. Equation (10) Equationmodel (10) iswhich the general modelthe which determines the mean extinction coefficient as of a linear is line. the general determines mean extinction coefficient as a linear function the function of the distance stagemethod. of the suggested method. As discussed Section distance ratio index in the ratio thirdindex stage in of the the third suggested As discussed in Section 2.4, the in volume 2.4, the volume temporal decorrelation reduces the coherence amplitude states in all polarization states but temporal decorrelation reduces the coherence amplitude in all polarization but it does not affect it does not affect the coherence phase at all. As a result, the radius segment beyond
γ tv γ v
up to the
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the coherence phase at all. As a result, the radius segment beyond |γtv |γv up to the CUC is the γv CUC is the γ vline ambiguous segment as shown in Figure 13 byred theline. dotted red line. Accordingly, ambiguous segment line as shown in Figure 13 by the dotted Accordingly, the corrected the corrected observed volume only coherence is on the ambiguous line segment of the observed volume only coherence is on the ambiguous line segment of the observed volumeobserved coherence. volume coherence. Furthermore, the fixed mean curve is locus. the volume coherence locus. Furthermore, the fixed mean extinction curve is theextinction volume coherence Therefore, the intersection
ϕ = ϕ v isobserved selected volume as the Therefore, the intersection between the and volume curve andcorrected point between the volumepoint coherence curve ϕ = coherence ϕv is selected as the coherence and the volume corresponding volume and the volume temporal decorrelation are the corrected observed coherence and height the corresponding volume height and the volume four-stage solution couple. temporal decorrelation are the four-stage solution couple.
Figure representation of of thethevolume Figure13. 13.Geometrical Geometrical representation volumeheight heightand andvolume volumetemporal temporaldecorrelation decorrelation estimation in the four-stage algorithm. estimation in the four-stage algorithm.
1. 1. 2. 2. 3. 3. 4.4.
In summary, the proposed four-stage method follows the following four steps. In summary, the proposed four-stage method follows the following four steps. Fit the least square line on the Pauli basis coherences and find its intersections with CUC as the Fit thecoherence least square line on the Pauli basis coherences and find its intersections with CUC as the ground candidates. groundthe coherence Choose ground candidates. coherence between two candidates according to the surface to volume Choose the ground backscattering ratio. coherence between two candidates according to the surface to volume backscattering Calculate the D.I ratio. using Equation (9) and estimate the mean extinction coefficient using Equation (10). Calculate the D.I using Equation (9) and estimate the mean extinction coefficient using Equation (10). Find coherencelocus locususing using Equation estimate the volume Findthe thefixed fixedmean mean extinction coherence Equation (4) (4) andand estimate the volume height height and the real temporal decorrelation multiplying factor from the intersection point and the real temporal decorrelation multiplying factor from the intersection point between the volumethe coherence and ϕ = . ϕ = ϕv . between volume loci coherence lociϕvand
3. Results 3. Results The coregistered SAR image pair was used to generate the coherency matrix using PolSARpro The coregistered SAR image pair was used to generate the coherency matrix using PolSARpro version 5.1 (https://earth.esa.int/web/polsarpro/home). First, the flat earth phase removal step was version 5.1 (https://earth.esa.int/web/polsarpro/home). First, the flat earth phase removal step was applied and then coherency matrices were calculated using an 11 × 11 boxcar filter. Figure 14 depicts applied and then coherency matrices were calculated using an 11 × 11 boxcar filter. Figure 14 depicts the efficiency of the proposed four-stage algorithm in the volume thickness estimation, in comparison the efficiency of the proposed four-stage algorithm in the volume thickness estimation, in comparison with the three-stage method for a pixel. The volume height has been estimated in a single pixel, with the three-stage method for a pixel. The volume height has been estimated in a single pixel, with with a height of 20.02 m, based on the RVoG and RVoG+VTD model using three-stage and four-stage a height of 20.02 m, based on the RVoG and RVoG+VTD model using three-stage and four-stage algorithms respectively. As already noticed, the first two steps of the mentioned methods are the algorithms respectively. As already noticed, the first two steps of the mentioned methods are the same. Therefore, the ground underlying phase has been estimated based on the first two stages of same. Therefore, the ground underlying phase has been estimated based on the first two stages of the the three-stage method. The last two parameters of the RVoG model (i.e., volume height and the three-stage method. The last two parameters of the RVoG model (i.e., volume height and the mean mean extinction coefficient); can be estimated at the final stage of the conventional three-stage method. extinction coefficient); can be estimated at the final stage of the conventional three-stage method. According to Equation (4), the volume height and the mean extinction coefficient corresponding According to Equation (4), the volume height and the mean extinction coefficient corresponding to to the observed volume coherence are the RVoG solution. Figure 14 depicts all fixed extinction the observed volume coherence are the RVoG solution. Figure 14 depicts all fixed extinction coherence curves corresponding to the mean extinction coefficient varied between 0 to 0.9 dB/m. The coherence curves corresponding to the mean extinction coefficient varied between 0 to 0.9 dB/m. The calculated volume coherence candidates are determined according to the fixed extinction coherence loci and the fitted coherence line intersection points, as shown in Table 1. Finally, the calculated
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calculated volume coherence candidates are determined according to the fixed extinction coherence loci and coherence the fitted coherence line intersection points, as shown in from Tablethe 1. Finally, thevolume calculated volume volume can be chosen based on the shortest distance observed coherence. coherence be chosen based onmean the shortest distance from the observed volume coherence. As shown can in Figure 14, the fixed extinction coherence locus corresponding to γ v (hv ,σ As = 0shown dB/m) in Figure 14, the fixed mean extinction coherence locus corresponding to γ h , σ = 0 dB/m ( ) v v is the nearest curve to the γ v . However, this is not physically possible as it generates negative isμ the in nearest curve to the γv . However, this is not physically possible as it generates negative µ in the γv . As a γ v γ. vAs selected fixed meancoherence extinction coherence locus in the the ) is the .1 dB/m v (hv , σ)= result, = 0.1γdB/m is0the selected fixed mean extinction locus in the conventional (hva, σresult, conventional three-stage method and the corresponding is 26 m.the Asvolume expected, the three-stage method and the corresponding volume heightvolume is 26 m.height As expected, height volume height is overestimated by using the RVoG model because this model interprets the temporal is overestimated by using the RVoG model because this model interprets the temporal decorrelation decorrelation as the volumetric decorrelation. at Table 1 reveals that the mean extinction as the volumetric decorrelation. A look at TableA1look reveals that the mean extinction coefficient is an coefficient is andecisive important and decisive inmethod. the three-stage Hence, the RVoG+VTD important and parameter in theparameter three-stage Hence,method. the RVoG+VTD inversion using inversion usingmethod the three-stage howisgood themean guessextinction is for the coefficient mean extinction the three-stage dependsmethod on howdepends good theon guess for the value. coefficient to the lack of information about the mean value,method the threeTherefore, value. due toTherefore, the lack ofdue information about the mean extinction value,extinction the three-stage is stage method is not to invert model. the RVoG+VTD model. not able to invert theable RVoG+VTD
Figure representation forfor thethe coherence locus selection usingusing three-stage and fourFigure 14. 14. The Thegeometrical geometrical representation coherence locus selection three-stage and stage algorithms. The blue curves are the coherence loci selected usingusing RVoG and four-stage algorithms. The and bluemagenta and magenta curves arefixed the fixed coherence loci selected RVoG RVoG+VTD models, respectively. The The dotted red line is the line line segment for γ and RVoG+VTD models, respectively. dotted red line is ambiguous the ambiguous segment for v .γv . Table 1. Table 1. Look-up Look-up table. table. Distance Value in CUC between and Distance Value in CUC between γv γand v γHVγ HV
MeanExtinction ExtinctionCoefficient Coefficient Mean
Estimated Forest Height Estimated Forest Height
0.0062 0.0062 0.1706 0.1706 0.2574 0.2574 0.3069 0.3069 0.3371 0.3371 0.3578 0.3578 0.3677 0.3677 0.3763 0.3763 0.3835 0.3835 0.3861 0.3861
00 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9
30 30 26 26 23 23 21 21 21 21 21 21 22 22 22 22 22 22 23 23
In this this respect, respect,the theproposed proposedfour-stage four-stagemethod method used forest height estimation based on is is used forfor forest height estimation based on the the RVoG+VTD model. As discussed in Section the mean extinction coefficient is estimated in RVoG+VTD model. As discussed in Section 2.5, 2.5, the mean extinction coefficient is estimated in the the third stage of the four-stage algorithm using Equations (9) and (10). As illustrated in Figure 14, third stage of the four-stage algorithm using Equations (9) and (10). As illustrated in Figure 14, the the A.L obtained 0.5175 and 1.2545for forthe theselected selectedpixel pixelwhich whichleads leads to to the the D.I == 0.4125 A.L andand V.LV.L areare obtained as as 0.5175 and 1.2545 based on based on Equation Equation (9). (9). Finally, the the mean mean extinction extinction coefficient coefficientisisestimated estimatedasas0.3 0.3dB/m dB/m based Equation (10). The fixed mean extinction volume curve corresponding to the γ h , σ = 0.3 dB/m is ( v γv (h , σ = 0.3 dB/m )) is Equation (10). The fixed mean extinction volume curve corresponding to the v v shown by the magenta curve in Figure 14. Finally, the intersection point between γ (h , σ = 0.3 dB/m) shown by the magenta curve in Figure 14. Finally, the intersection point between vγ v (vhv ,σ = 0.3 dB/m) and ϕ = ϕv is selected as the corrected observed volume coherence and the corresponding volume and ϕ = ϕ v is selected as the corrected observed volume coherence and the corresponding volume
height, hv = 18 m , is the estimated volume thickness using the four-stage algorithm. As a result, the estimated heights using the three-stage method and the proposed algorithm show that the
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height, hv = 18 m, is the estimated volume thickness using the four-stage algorithm. As a result, the Remote Sens. 2018, 10, x FOR PEER REVIEW 14 of 16 estimated heights using the three-stage method and the proposed algorithm show that the RVoG+VTD model effectively theremoves temporalthe decorrelation effect and the suggested provides a RVoG+VTD modelremoves effectively temporal decorrelation effect and themethod suggested method more accurate estimate of the forest height. provides a more accurate estimate of the forest height. Figure forest height Figure 15 15 demonstrates demonstrates the the estimated estimated forest height using using 15 15 selected selected stands stands versus versus the the LiDAR LiDAR reference height. The real data experimental results show that the RVoG+VTD model effectively reference height. The real data experimental results show that the RVoG+VTD model effectively removes temporal decorrelation decorrelation effect effect and and provides provides aa more more accurate accurate estimate estimate of of the the forest forest height. height. removes the the temporal The The root root mean mean square square error error (RMSE) (RMSE) indicates indicates 5.42 5.42 m m improvement improvement compared compared to to the the three-stage three-stage method. Alternatively, the fit of the RVoG+VTD model and the efficiency of the proposed method. Alternatively, the fit of the RVoG+VTD model and the efficiency of the proposed method method 2 In this case, the value of the determination of were were checked checked by by the the determination determination of of coefficient, coefficient, R R2.. In this case, the value of the determination of 2 coefficient 0.8121) indicates indicates that that the theRVoG+VTD RVoG+VTD inversion inversion using using the the four-stage four-stage method method provides provides coefficient (R (R2 ==0.8121) an efficiency of over 81%. an efficiency of over 81%.
Figure 15. Comparison between LiDAR measurements and forest height inversion from three-stage and four-stage algorithms.
4. Discussion The proposed four-stage algorithm extends the conventional three-stage method to invert the RVoG+VTD model without the need for any auxiliary data. In other words, the suggested inversion method takes advantages of desirable characteristics of the three-stage method, while avoiding its shortcomings. As As shown shown in in Figure Figure 8, 8, as as the forest height increases, the coherence loci fall into the origin of ofthe thecomplex complex unit circle. Accordingly, the RVoG overestimates theheight forestbecause height unit circle. Accordingly, the RVoG modelmodel overestimates the forest because thisinterprets model interprets the temporal decorrelation as the volumetric decorrelation. the this model the temporal decorrelation as the volumetric decorrelation. Thus, the Thus, temporal temporal decorrelation effects is incorporated in the RVoG+VTD model by definingmultiplying a volume decorrelation effects is incorporated in the RVoG+VTD model by defining a volume multiplying factor. A extinction fixed meancoefficient extinctioncan coefficient be used to obtain unique volume height factor. A fixed mean be usedcan to obtain unique volume height estimation estimation in the of the temporal decorrelation [10]. In the thisproposed respect, the proposed in the presence of presence the temporal decorrelation effect [10]. Ineffect this respect, method used method usedratio the index distance ratio index to calculate reliablecoefficient. mean extinction coefficient. The the distance to calculate the reliable meanthe extinction The proposed method, proposed method, estimates the forest height more than due the three-stage due todecorrelation the temporal estimates the forest height more accurate than theaccurate three-stage to the temporal decorrelation compensation using themodel. RVoG+VTD model. proposed method solves the compensation using the RVoG+VTD Moreover, theMoreover, proposedthe method solves the RVoG+VTD RVoG+VTD model using only a single-baseline PolInSAR data. As shown in Figure 15, the results of model using only a single-baseline PolInSAR data. As shown in Figure 15, the results of the RVoG the RVoGusing inversion using three-stage method from aerror, systematic error, which is decorrelation the temporal inversion three-stage method suffer from asuffer systematic which is the temporal decorrelation effect. Inthe this respect, the RVoG using repeat pass to the effect. In this respect, RVoG inversion usinginversion repeat pass PolInSAR data PolInSAR leads to thedata biasleads estimation bias estimation and the RVoG+VTD inversion using four-stage method reduces the bias by defining and the RVoG+VTD inversion using four-stage method reduces the bias by defining a temporal a temporal decorrelation factor. decorrelation factor. Some limitations of the four-stage method are also highlighted. It should be considered that the vertical homogenous layer with a constant extinction coefficient along the vertical direction is not reliable if the dominant scatterers are not vertically uniformly distributed in the canopy [25]. In this respect, the efficiency of the proposed method can be investigated in the forest height estimation using vertically varying extinction.
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reliable if the dominant scatterers are not vertically uniformly distributed in the canopy [25]. In this respect, the efficiency of the proposed method can be investigated in the forest height estimation using vertically varying extinction. 5. Conclusions The temporal baseline limits the capability of the interferometric coherence over volumetric media. The temporal decorrelation is defined as a real valued multiplying factor in the RVoG+VTD model. The new parameter, |γtv |, makes the RVoG+VTD inversion, in terms of a single-baseline fully-polarimetric configuration, a challenge. Probably the simplest way to overcome this ambiguity is to set the mean extinction coefficient to a fixed value. However, the accuracy of results depends on how well the mean extinction value is chosen. In this paper, we proposed a new geometrical method to estimate the volume height and the volume temporal decorrelation using the RVoG+VTD model. The proposed method used a geometrical index to estimate the mean extinction coefficient in an independent stage. In conclusion, the proposed four-stage algorithm inverts the RVoG+VTD model without the need for a prior information about temporal decorrelation multiplying factor or mean extinction coefficient. The proposed algorithm improves the accuracy and reduces the bias effectively so that the root mean square error, RMSE, indicates a significant improvement compared to the RVoG inversion using the conventional three-stage method. Author Contributions: Conceptualization, T.M. and Y.M.; Methodology, T.M.; Project administration, Y.M.; Software, T.M.; Supervision, Y.M. and M.J.V.Z.; Validation, T.M.; Formal Analysis, T.M., Y.M. and M.J.V.Z.; Investigation, T.M., Y.M. and M.J.V.Z., Writing-original draft Preparation, T.M.; Writing-Review & Editing, T.M., Y.M. and M.J.V.Z. Funding: This research received no external funding. The APC was funded by K. N. Toosi University of Technology. Acknowledgments: The authors acknowledge the European Space Agency for providing valuable images and field data in the BioSAR 2007 campaign. Conflicts of Interest: The authors declare no conflict of interest.
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