Remote Sensing of Environment 193 (2017) 29–37

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Remote Sensing of Environment

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Polarized reflectances of urban areas: Analysis and models

Donghai Xie a,c, Tianhai Cheng b,⁎, Yu Wu b, Han Fu a,c, Ruofei Zhong a,c, Jie Yu a,c

a State Key Laboratory Incubation Base of Urban Environmental Processes and Digital Simulation, Capital Normal University, Beijing, Chinab Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing, Chinac Beijing Key Laboratory of Resource Environment and Geographic Information System, Capital Normal University, Beijing, China

⁎ Corresponding author.E-mail address: cthy@irsa.ac.cn (T. Cheng).

http://dx.doi.org/10.1016/j.rse.2017.02.0260034-4257/© 2017 Elsevier Inc. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 6 October 2016Received in revised form 9 February 2017Accepted 28 February 2017

Accurate BPDF (Bidirectional Polarization Distribution Functions) of urban will help to improve the accuracy ofinverted aerosol parameters that is very important for the research of urban atmospheric pollution and climatechange. With Fresnel formula of polarization as the foundation, this paper studies three important factors thatinfluence the BPDF: shadow, slope distribution and NDVI, and proposes a new BPDF model (named as Xie-Chengmodel for convenience in comparison) for urban areas. Because the influence of slope distribution is trivial,only two parameters are needed in the newmodel, one controlling the shadow and the other for overall scale. Anexperimental factor is introduced into the model to compensate the influence of NDVI for polarized reflectancemore accurately. Experiments prove that new model performs best in both correlation and RMSE for measure-ments not only with different urban places around the world but also clustered urban data by different NDVIs.Compared with the best current BPDF model, new model can reduce the average RMSE error by about 4.5% fordifferent urban areas. Error distribution in polar coordinates also shows that newmodel can achieve smallest er-rors in almost all directions under fixed sun zenith angle.

© 2017 Elsevier Inc. All rights reserved.

Keywords:BPDFUrban areasPARASOLFresnel equationNDVI

1. Introduction

Due to the unique advantages of polarization, polarized remotesensing using multiple-viewing-angle polarized measurements hasbeen widely used in the retrieval of aerosol properties (Deuzé et al.,2001; Herman, 2005; Waquet et al., 2009a; Tanré et al., 2011; Duboviket al., 2011; Cheng et al., 2012; Xie et al., 2013; Kokhanovsky et al.,2015). One of the key factors influencing the precision of aerosol re-trieval is polarized reflectance of surface in different directions, whichcan be modeled by BPDF (bidirectional polarized distribution function).Compared with the surface non-polarized reflectance, surface BPDF isusually considered as spectrally independent (Waquet et al., 2009b) inthe visible and infrared regions and described by models based on theassumption of single Fresnel reflection from the surface facets(Roujean et al., 1992; Bréon et al., 1995; Nadal and Bréon, 1999;Maignan et al., 2009; Waquet et al., 2009b; Litvinov et al., 2011, 2012;Xiang et al., 2015, 2016). Although many BPDF models had been devel-oped, none of them was about urban areas, over which the aerosol iscrucial for climate change research (Hansen et al., 2005).

Roujean et al. (1992) presented an analysis of polarizationmeasurements acquired over corn and soybean crop canopies and de-veloped a simple physical BPDF model based on the hypothesis thatleaves specularly reflect light according to the Fresnel equations.

Without considering any attenuation on the incident and outgoingpath, Bréon et al. (1995) developed a physically based BPDF for baresoil assuming that the ground is composed of isotropically distributedfacets (rough surface). Based on spaceborne POLDER (Polarization andDirectionality of Earth's Reflectances) polarization measurements oftwo months, Nadal and Bréon (1999) proposed a semi-empirical sur-face BPDF that was adopted by POLDER aerosol product processingline (Deuzé et al., 2001).

Waquet et al. (2009b) compared four BPDFmodels including scaledFresnel model, linear combination model developed for bare soil andvegetation (Bréon et al., 1995), Nadal-Bréon model (Nadal and Bréon,1999), and scaled Fresnel model with shading factor validated byMICROPOL measurements for closely cropped surfaces and forest intheNorth of France. The results confirmed that the polarization generat-ed by the reflection of vegetated surfaces could be considered as beingprimarily a specular reflection process. Comparison showed that the lin-ear combinationmodel is not adequate tofit the observed surface polar-ization angular behavior and Nadal-Bréon model would overestimatethe surface polarized reflectance for closely cropped surfaces. Maignanet al. (2009) had performed an extensive comparison of differentBPDF models with POLDER satellite data and introduced a new one-parametric model that allowed a similar fit to POLDER data as a previ-ously developed Nadal-Bréon model (Nadal and Bréon, 1999).

Litvinov et al. (2011) proposed a three-parameter BPDF model andtested different models of the BPDF for bare soil and vegetation surfacesusing multi-angle, multi-spectral photo polarimetric airborne

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measurements of the Research Scanning Polarimeter (RSP) which was aprototype for the Aerosol Polarimetry Sensor instrument of the NASAGlory Project (Cairns et al., 1999; Mishchenko et al., 2007). Experimentsshowed that the new BPDF model was better than the semi-empiricalNadal-Bréon model (Nadal and Bréon, 1999) and the model developedby Maignan et al. (2009). Litvinov et al. (2012) derived physical BPDFmodel from the general solution of the electromagnetic scattering prob-lems by random media. Multi-angle photo polarimetric airborne mea-surements of the RSP and satellite POLDER measurements were used toinvestigate the performance of the presented model.

Compared with the simple land cover types, such as forest, grass-land, cropland and desert et al., the combination of urban is very com-plex, including many buildings, trees and roads. This paper proposes anew semi-empirical BPDFmodel for urban areas. Themulti-angle polar-ized data contained in PARASOL (Polarization and Anisotropy of Reflec-tances for Atmospheric science coupledwith Observations from a Lidar)BRDF are used to test our new model and achieve good approximationcompared with existing models. Only polarized reflectance of 865 nmis used in this paper and the surface-polarized reflectance was consid-ered to be spectrally neutral (Nadal and Bréon, 1999), but in factmany objects in urban areasmay be spectrally invariant such as plasticsand coated roof, which is beyond the scope of this article and should bea potential issue of urban areas using new instruments withmore chan-nels, higher precision and resolution such as 3MI (Marbach and Riedi,2015) in the future.

This paper is organized as follows. Section 2 describes the data anddefinitions for our experiments. Section 3 compares the state-of-artBPDFmodels and proposes our newmodel for urban areas. The compar-isons of different BPDF models in terms of correlation and RMSE arediscussed in Section 4. Finally, conclusions are given in Section 5.

2. Data and definitions

The urban data used in this paper are from BRDF databases generat-ed from POLDER instrument onboard the PARASOL satellite. Althoughthe databasewas named by BRDF andmainly concerned the directionalsignatures of the reflectance, it was also extended to provide the polar-ized reflectance at 865 nm (near infrared) which is much less affectedby the highly polarized molecular and aerosol scattering than theshorter wavelength channels, such as 490 nm and 670 nm measuredby POLDER instrument. In-flight calibration of polarized channels wascarried out using the sun's glitter and the expected accuracy is about0.5% in the near-infrared channel 865 nm and about 2% in the visiblechannels, in terms of percent polarization (Toubbe et al., 1999). TheLSCE, one of the POSTEL Expertise Centre, defined a new method to se-lect the BRDFs fromPARASOL data acquired fromNovember 2005 to Oc-tober 2006 in order to build four BRDF databases: two monthlydatabases gathering the best quality BRDFs for each month indepen-dently, two yearly databases designed to monitor the annual cycle ofsurface reflectance and its directional signature whose selection ofhigh quality pixels was based on the full year. The monthly and yearlyBRDF databases were organized based on the IGBP classification andthe GLC2000 land cover map. In this paper, we use the urban type(IGBP class: 13) data from BRDF database classified by IGBP land covermap which can be downloaded freely from POSTEL website (http://postel.mediasfrance.org).

Table 1Comparison of previous BPDF models.

Authors Year Type Num. of parameters

Rondeaux et al. 1991 Physical 0Bréon et al. 1995 Physical 0Nadal & Bréon 1999 Semi-empirical 2Waquet et al. 2009 Semi-empirical 2Maignan et al. 2009 Semi-empirical 1Litvinov et al. 2011 Semi-empirical 3

In the BRDF database, the multi-angle reflectances of 6 bands (490,565, 670, 765, 865, 1020 nm) and polarized reflectance of 865 nm,along with the geometrical angles of each measurement direction(sun zenith, view zenith, and relative azimuth) of one month for eachsite were saved into a file. Only linear polarization was considered inthe BRDF database, so the polarized reflectance Rp can be defined as:

Rp ¼ πffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQ2 þ U2

qE0 cos θsð Þ ð1Þ

where Q and U are the components of Stokes vector, θs is the sun zenith,E0 is the irradiance at the top of atmosphere. All fileswere firstly dividedinto different directories according to the classification and then orga-nized by NDVI, which is an important parameter to distinguish differentsurface cover types. The average NDVI of onemonthwas calculated andsaved into the file.

3. BPDF model for urban type

Several BPDFmodels were proposed up to now, which can be divid-ed into two categories: physical and semi-empirical. But none of themwas developed and validated for urban type specially. Table 1 lists allthe previous BPDF models (see Appendix B) and compares their differ-ences in several aspects: type, number of parameters, surface types anddevice.

Based on the principle of polarized reflectance and characteristics ofurban areas, we propose a new BPDF model for urban type

Rp ¼ A� Fp N;γð Þ � f sh γð Þ � exp −w� NDVIð Þ ð2Þ

f sh γð Þ ¼ 1þ coskγ π−γð Þ2

� �3

ð3Þ

where w=0.7 and two parameters are included in new model: A andkγ. Fp(N,γ) is the polarized Fresnel function to describe the polarizedspecular reflectance (see Appendix A). The shadowing function fsh(γ)was used in the Litvinov model and kγ controls the shadowing. Inorder to compensate for the influence of NDVI to polarized reflectancebetter, we add an experimental parameter w. The physical model(Rondeaux and Herman, 1991) is not adopted as the foundation of ournew model like Nadal-Bréon, Maignan and Litvinov model, whereaswe use the polarized Fresnel function directly. The reason is that alarge amount of vegetation and bare soil are contained in the urbanareas, so the physical model designed for vegetation is not suitable forthe urban type. Our new model considers the influence of shadow andNDVI, whereas we do not introduce the slope function.

3.1. Shadow

According to the real measurements, the polarized reflectance de-creases with the scattering angle. The simulated polarized reflectancesof two physical models coincide with the measurements well whenthe scattering angle is large, but overestimate the measurements asthe scattering angle is smaller than some threshold. This phenomenoncan be explained by shadowing effect. When seen from the backwardscattering direction (scattering angle = 180°), there are no places

Surface types Device

Crop canopies Polarimeter from LOABare soils, vegetaion REFPOLForest, shrublands, low vegetation, desert POLDERForest, closely cropped surfaces MICROPOL14 IGBP classes except for urban PARASOLSoil, vegetation RSP

Table 2Ten locations from yearly database.

Site ID Directions Months Lat, Lon (degree) Country

S1 1591 12 32.75, 51.49 IranS2 1673 12 14.81, 42.96 YemenS3 1346 12 36.03, −114.97 USAS4 1700 12 29.14, 58.36 IranS5 1438 11 33.92, −117.24 USAS6 1397 11 33.36, −111.92 USAS7 1103 12 41.19, −8.60 PortugalS8 1314 11 34.03, −117.75 USAS9 1076 11 43.69, 3.88 FranceS10 1014 10 41.47, −8.34 Portugal

Fig. 1. Correction for attenuation of polarized reflectance caused by NDVI using exp(−w∗NDVI), left: w=1, right: w=0.7.

31D. Xie et al. / Remote Sensing of Environment 193 (2017) 29–37

obscured by other objects. But the shadowing areas increase when wesee the objects from the forward direction. Shadowing areaswill not re-flect the light from sun, so the polarized reflectance will decrease. Tosimply the mathematical deduction, the physical vegetation and soilmodel do not consider the mutual occluding of different surface targetsand shadowing effect is ignored, so they will overestimate the real po-larized reflectances for small scattering angles. There are three BPDFmodels (Litvinov et al., 2011; Maignan et al., 2009; Waquet et al.,2009b) having shadowing function in their models, the one in Maignanmodel accounted for thepresence of vertical protuberancewhich is sim-ilar to the shadow. The shadowing part of Waquet model is related to

Fig. 2. Distribution of ten sites marke

the zenith angles (Saunders, 1967), the other two are based on scatter-ing angle or phase angle (phase angle=(180° – scattering angle) / 2). Inthis paper, we adopt the shadowing function from Litvinov model(Litvinov et al., 2011).

3.2. NDVI

NDVI is an important index in remote sensing which can be used todiscriminate typical vegetation, soil, or water cover types. The NDVI ofvegetation is always higher than soil, whereas the NDVI of water is al-ways negative. According to experimental measurements, the surfaceNDVI is in inverse proportion to the polarized reflectance because thehigher NDVI surfaces include more vegetation whose polarized reflec-tance is lower than soil. In themodel of Nadal, all IGBP types are dividedinto four classes and each class is then divided byNDVI. In all themodelslisted in Table 4, NDVI is considered only inMaignanmodel. Urban areasoften include many vegetation cover types, such as grasses, trees,shrubs, et al. So the BPDF model for urban must consider the influenceof vegetation growing, which is often reflected by NDVI. In the deriva-tion of Maignan model, the empirical function obtaining the best resultis exp(−NDVI). But according to our analysis, exp(−w∗NDVI) with anempirical parameter w will compensate the attenuation better. Inorder to obtain the best value of w, we collected all polarized measure-ments whose scattering angles lie within 100∘±1∘ from the BRDF data-base. Different values of wwere used to correct the measurements and

d by red stars on the world map.

Fig. 3. Polarized reflectances of ten sites from from November 2005 to October 2006 in PARASOL BRDF database. S1,S2, …,S10 are the site IDs in Table 2.

32 D. Xie et al. / Remote Sensing of Environment 193 (2017) 29–37

Fig. 4.Measurements and modeling of three classes of NDVI for five BPDF models.

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then execute linear fitting. As shown in Fig.1, the slope of linear fitting isclose to zero when w=0.7.

3.3. Slope distribution

Earth surface is composed with many tiny objects like leaf surfacesor soil facets whose reflectances can be modeled by specular reflectiondescribed by Fresnel function. As shown by Fresnel function, the polar-ized reflectance is only decided by refractive index and scattering angle,which can be calculated by incident and reflection direction (sun zenithand azimuth, view zenith and azimuth). Once the scattering angle isfixed, the facets slopes determine the amount of reflection. The Gauss-ian function is often used to simulate the distribution of slope(Litvinov et al., 2011). In order to analyze the influence of slope distribu-tion to the urban polarized reflectance, we use all 1074 urban type filesfrom PARASOL monthly BRDF based on IGBP and calculate the RMSEerror of our new model in two circumstances: with and without slopedistribution. The RMSE with slop distribution is 0.002043 and RMSEwithout slope distribution is 0.002059. The difference is so trivial thatwe can ignore the influence of slope for urban type.

4. Experiments and analysis

We use the files of urban type from BRDF yearly database of PARA-SOL classified by IGBP to conduct experiments and compare the perfor-mances of BPDF models. Previous experiments had proven thatRondeaux model strongly underestimates the polarization, and thetwo physical BPDFmodels for vegetation and soil lead to large overesti-mates for the large sun and view angles. Sowe only compare fivemodesnamed as Nadal-Bréon,Waquet, Maignan, Litivinov and Xie-Cheng (ournew model). Three strategies are designed to verify the accuracy ofthese models: first for data distributed in different places, second for

Table 3Statistics on the results of the Nadal-Bréon model, Waquet model, Maignan model, Litvinov modatabase.

Site ID Nadal-Bréon model Waquet Model Maigna

cor. RMSE × 100 cor. RMSE × 100 cor.

S1 0.95 0.15(0.10) 0.88 0.27(0.17) 0.88S2 0.95 0.22(0.15) 0.95 0.22(0.15) 0.95S3 0.91 0.21(0.13) 0.92 0.23(0.17) 0.92S4 0.98 0.12(0.09) 0.96 0.22(0.16) 0.96S5 0.92 0.26(0.15) 0.93 0.25(0.16) 0.93S6 0.93 0.24(0.15) 0.95 0.20(0.13) 0.95S7 0.90 0.24(0.15) 0.87 0.30(0.20) 0.87S8 0.93 0.27(0.16) 0.94 0.25(0.15) 0.93S9 0.86 0.21(0.14) 0.84 0.25(0.18) 0.84S10 0.87 0.29(0.17) 0.87 0.31(0.20) 0.87Ave 0.92 0.22(0.14) 0.91 0.25(0.17) 0.91

The best statistics have italic bold characters. The average absolute difference multiplied by 10

data based on NDVI classification, third for data under fixed sun zenithangle.

4.1. Urban data from different sites around the world

In order to compare the performance of the BPDF models for urbanpolarized reflectance in different places, we collect the data of tenurban targets from the BRDF database based upon IGBP. Those targetswere seen clearly by PARASOL satellite clearly at least 10 months fromNovember 2005 to October 2006. Detailed information of the ten sitesis listed in Table 2.

In Table 2, the ten sites are named by Site ID from S1 to S10, and thenumber of measurements is listed in column “Directions”. The column“Months” records the number of months when the urban sites can beseen clearly by PARASOL satellite. We also display the locations of tensites in world map in Fig.2.

As shown in the Fig.2, four urban sites are distributed inNorth Amer-ica, three are in Europe and the other three sites are from Asia. All tensites are located in North Hemisphere. In order to observe the polarizedreflectance and compare their difference directly, we display all the po-larized reflectances along the scattering angle in Fig.3.

From Fig.3, we can see that polarized reflectances of different siteshave obvious differences, whereas thedistribution ismore concentratedthan the clustered data according to the NDVI (Fig.4). All polarized re-flectances decrease as the scattering angle increases. The directions ofall measurement data are input into the models and the parametersare calculated based on nonlinear optimization. The statistics on the re-sults of all models are listed in Table 3.

As shown in Table 3, Xie-Chengmodel can reduce the average RMSEerror by about 4.5% compared with the Nadal-Bréonmodel which is thebest current model. Waquet and Maignan model are worst in fivemodels, especially the average RMSE is about 20% higher than Xie-

del and Xie-Cheng model (correlation and RMSE) for 10 urban sites from PARASOL BRDF

n model Litvinov model Xie-Cheng model

RMSE × 100 cor. RMSE × 100 cor. RMSE × 100

0.16(0.11) 0.94 0.17(0.12) 0.94 0.17(0.11)0.25(0.16) 0.94 0.24(0.15) 0.95 0.22(0.14)0.24(0.15) 0.91 0.22(0.14) 0.92 0.20(0.13)0.13(0.10) 0.97 0.14(0.10) 0.98 0.12(0.09)0.30(0.17) 0.91 0.26(0.16) 0.93 0.24(0.13)0.30(0.17) 0.93 0.24(0.14) 0.95 0.21(0.12)0.25(0.16) 0.90 0.25(0.16) 0.91 0.24(0.14)0.37(0.20) 0.91 0.28(0.16) 0.94 0.26(0.14)0.20(0.14) 0.86 0.22(0.15) 0.87 0.20(0.14)0.32(0.18) 0.85 0.32(0.19) 0.88 0.28(0.17)0.25(0.15) 0.91 0.23(0.15) 0.93 0.21(0.13)

0 is listed in parentheses in the columns named by RMSE × 100.

Table 4BPDF model parameters of five models for 10 sites.

Site ID Nadal-Bréonmodel

Waquetmodel

Maignanmodel

Litvinov model Xie-Chengmodel

α β ξ σ C α σ kγ A kγ

S1 0.019 56.69 0.57 1.18 5.71 3.17 0.62 0.66 0.76 0.59S2 0.047 27.58 0.73 0.62 8.86 3.34 0.54 0.52 0.85 0.36S3 0.017 62.66 0.70 1.64 5.94 2.78 0.51 0.71 0.74 0.56S4 0.027 49.23 0.75 1.19 8.04 3.61 0.57 0.64 0.93 0.53S5 0.028 34.19 0.60 1.00 7.85 2.64 0.53 0.59 0.72 0.39S6 0.028 34.37 0.63 1.03 7.70 2.65 0.52 0.60 0.72 0.38S7 0.018 51.68 0.52 1.07 7.06 2.92 0.58 0.68 0.81 0.54S8 0.040 20.95 0.52 0.65 9.00 2.32 0.54 0.51 0.66 0.19S9 0.014 45.80 0.33 0.91 5.58 2.00 0.61 0.65 0.60 0.53S10 0.021 40.05 0.51 1.07 8.33 2.42 0.56 0.64 0.83 0.49

Table 6BPDF model parameters of five models for different NDVIs.

NDVI Nadal-Bréonmodel

Waquetmodel

Maignanmodel

Litvinov model Xie-Chengmodel

α β ξ σ C α σ kγ A kγ

0–0.1 0.021 64.97 0.72 1.30 6.99 3.59 0.57 0.68 0.89 0.580.1–0.2 0.023 51.15 0.59 0.97 7.21 3.30 0.59 0.64 0.82 0.520.2–0.3 0.026 44.43 0.55 0.80 8.31 3.26 0.59 0.62 0.88 0.500.3–0.4 0.022 46.19 0.54 0.99 7.88 2.88 0.57 0.64 0.83 0.510.4–0.5 0.022 41.54 0.50 0.94 8.20 2.66 0.56 0.63 0.83 0.490.5–0.6 0.026 28.44 0.42 0.69 8.44 2.10 0.56 0.56 0.69 0.370.6–0.7 0.014 61.37 0.565 1.62 8.15 2.36 0.54 0.70 0.90 0.59

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Cheng model. The average absolute difference is listed in parenthesesfor the sake of aerosol optical depth retrieval. Parameters of fivemodelsfor 10 sites are listed in Table 4.

4.2. Urban data classified by NDVI

We divide all files of urban type into seven classes according to theirNDVI. The measurements and models of three classes are plotted inFig.4.

From Fig.4, we can see that there are differences between the polar-ized reflectances of urban sites in different NDVI. Due to the complexityof target surfaces from different cities, the actual observations of polar-ized reflectances vary greatly.WhenNDVI is small or large, polarized re-flectance distribution is relatively concentrated.When the NDVI value isin themiddle, the variation in polarized reflectance is bigger. The reasonis that surface types are simple with small or large NDVI, but themiddleNDVI often means a complex combination in the urban sites. Due to thediversity of buildings and vegetation, the polarized reflectance variesgreatly.

From Table 5, we can see that Xie-Cheng model and Nadal-Bréonmodel perform best in correlation and RMSE. Litvinov model achievesthe same average correlation as Xie-Cheng model and Nadal-Bréonmodel. Just like Table 3, the average absolute difference is listed in pa-rentheses. Parameters of five models for different NDVIs are listed inTable 6.

4.3. Urban data under fixed sun zenith angle

In order to compare the error distributions of five models, we select375 directions from site S5 whose sun zenith angles lie from 20° to 30°and calculate the absolute error of each direction for five BPDF models.The error distribution is interpolated and plotted in polar coordinateswhose radius represents view zenith angle (shown in Fig.5).

From Fig. 5, we can see that the largest errors of five models appearin the forward scattering directionswhose scattering angle is small. Thereason is that polarized reflectance increases with the decrease of

Table 5Statistics on the results of the Nadal-Bréon model, Waquet model, Maignan model, Litvinovmo

NDVI Nadal-Bréon model Waquet model Maigna

corr. RMSE × 100 corr. RMSE × 100 corr.

0–0.1 0.87 0.29(0.18) 0.83 0.35(0.23) 0.840.1–0.2 0.89 0.26(0.16) 0.86 0.32(0.20) 0.850.2–0.3 0.83 0.42(0.25) 0.81 0.45(0.28) 0.830.3–0.4 0.81 0.38(0.23) 0.80 0.41(0.25) 0.810.4–0.5 0.79 0.43(0.25) 0.78 0.45(0.27) 0.770.5–0.6 0.84 0.32(0.18) 0.85 0.32(0.19) 0.840.6–0.7 0.78 0.29(0.18) 0.76 0.31(0.20) 0.76Ave 0.83 0.34(0.20) 0.81 0.37(0.23) 0.82

The best statistics have italic bold characters. The average absolute difference multiplied by 10

scattering angle. When the view zenith angle is larger, the errors of allmodels are also larger. The error difference of five models in all direc-tions is not obvious and themaximum value is less than 0.3%. It is inter-esting that the errors around the specular reflection are large for all fivemodels. The error of Xie-Cheng model in the forward scattering direc-tion of principal plane is higher than Maignan model and nearly thesame as the other three models. But in most other directions, Xie-Cheng model performs best especially for those backscattering direc-tions whose view zenith angles are large. So the average error of Xie-Cheng model is smallest on the whole.

5. Conclusion

Although many BPDF models were proposed in the past twentyyears and evaluated by measurements from field experiments or satel-lites, none of them was developed for urban areas specially. This paperanalyzed the characteristics of polarized reflectance of urban areasand proposed a new BPDF model (Xie-Cheng model) for urban type.We analyzed three important factors in BPDF models: shadow, NDVIand slope and found that NDVI must be considered by an experimentalparameters, but the slope distribution function can be discarded basedon the accuracy comparison. Of the three factors, shadow is the oneinfluencing the BPDF mostly. Considering the complex components ofurban areas, we adopted the Fresnel function as the core part of ournew model directly rather than the physical model for vegetation(Roujean et al., 1992).

We used those polarized reflectances of urban areas from PARASOLBRDF database to test our new BPDF model. In the first experiment,we selected 10 sites distributed around the world and calculated theRMSE and correlation of all five models. Our new model performs bestin all models and the RMSE is about 4.5% lower than the Nadal-Bréonmodel, which is lowest among the four previous models. In the secondexperiment, we classified the polarized data according to the NDVIand also calculated the RMSE and correlation of different models andour newmodel performs best. To compare the error distribution in dif-ferent directions, we selected all directions from site S5 whose sun ze-nith angles lie in a small range of sun zenith and calculated theabsolute error of each direction for five BPDFmodels in the third exper-iment. The error distribution was interpolated and plotted in polar

del and Xie-Cheng model (correlation and RMSE) for different NDVIs from BRDF database.

n model Litvinov model Xie-Cheng model

RMSE × 100 corr. RMSE × 100 corr. RMSE × 100

0.30(0.19) 0.87 0.30(0.18) 0.87 0.29(0.17)0.26(0.16) 0.89 0.26(0.16) 0.89 0.26(0.16)0.42(0.25) 0.82 0.43(0.26) 0.82 0.42(0.25)0.39(0.23) 0.80 0.40(0.24) 0.81 0.38(0.22)0.44(0.25) 0.77 0.45(0.26) 0.79 0.43(0.24)0.34(0.19) 0.83 0.34(0.19) 0.85 0.32(0.18)0.30(0.18) 0.78 0.29(0.18) 0.79 0.29(0.17)0.35(0.21) 0.83 0.35(0.21) 0.83 0.34(0.20)

0 is listed in parentheses in the columns named by RMSE × 100.

Fig. 5. Interpolated error distributions of five models in polar coordinates. The circle represents the average sun position whose zenith angle is 24° and azimuth angle is 0°.

35D. Xie et al. / Remote Sensing of Environment 193 (2017) 29–37

coordinates. Except for the forward scattering direction in which ourerror is higher than Maignan model, our new model performs best inother directions, especially for those back scattering directions whoseview zenith angles are large.

The polarized calibration accuracy of POLDER/PARASOL was about0.5% for 865 nm in polarization degree, so the level of accuracy of themeasurementswill be lower than 0.001 in RMSE, whichmakes it neces-sary to develop more precise BPDF model just as this paper does. BPDFmodel was mainly used in the retrieval of aerosol optical depth and re-search showed that a typical error of 0.001 for the surface polarized re-flectance leads to an error on the aerosol optical depth of 0.04 (Maignanet al., 2009). According to our analyses, the average difference betweenthose previousmodels and our newmodel is less than 0.001, so our newmodel cannot improve the AOD accuracy obviously compared to theprevious four BPDF models listed in our paper. But polarimetric reflec-tivity is sensitive to the microphysical properties of aerosols. When wewant to retrieve the AODandmicrophysical properties of aerosol simul-taneously, high precise BPDF model will play an important role, inwhich higher precise polarized reflectance may distinguish differentaerosol conditions. So we think our new model is more meaningful toremote sensing of complete aerosol properties (Dubovik et al., 2011).

Because the urban areas consist of complex vegetation and build-ings, the combination of weighted vegetation and soil model is a goodchoice. But according to our experiments, there are negative weightsafter the optimization based on non-linear least squares, which also ap-prove the theory that vegetation and soil model are highly correlateddue to their common polarized Fresnel and attenuation terms(Maignan et al., 2009). Our new model can be used to the polarized re-mote sensing of aerosol properties over urban areas because of its sim-pleness, high precision and adaption to the variation in NDVI.

Acknowledgments

This work was supported by the National Natural Science Founda-tion of China (41371015, 41401386, 41501401, 41001207, 41671417),and the Major Special Project - the China High-Resolution Earth Obser-vation System (30-Y20A21-9003-15/17). The products used in this

paper were derived from the PARASOL instrument onboard the CNES/PARASOL satellite.We thank the ICAREData and Services Center for pro-viding data access and computing facilities necessary to perform thiswork. The Visu_BRDF_PARASOL tool developers are also acknowledgedfor providing a tool for visualizing the PARASOL BRDF databases.

Appendix A

In this appendix, we introduce the polarized Fresnel formula. Thepolarized reflectance is mostly generated by single specular reflectioneither on the leaf surface or on the soil facets, which is controlled byFresnel equations. Polarized fraction Fp(N,γ) of Fresnel reflectance is afunction of refractive index of reflective medium N and reflectionangle γ.

Fp N;γð Þ ¼ 12

r2⊥−r2¼� � ðA1Þ

r⊥ N;γð Þ ¼ NμT−μ I

NμT þ μ IðA2Þ

r¼ N;γð Þ ¼ Nμ I−μT

Nμ I þ μTðA3Þ

μ I ¼ cosγ;uT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−

sin2γN2

sðA4Þ

γ ¼ 12

cosθs cosθv þ sinθs sinθv cosφð Þ ðA5Þ

where r⊥(N,γ) and r=(N,γ) are respectively the perpendicular and par-allel Fresnel reflection coefficients, θs and θv are the zenith angle of solarand view, φ is the relative azimuth between the solar and view direc-tions. The refractive index of air is approximated to 1. The commonly ac-cepted value of refractive index for vegetation and bare soil is 1.5.

36 D. Xie et al. / Remote Sensing of Environment 193 (2017) 29–37

Appendix B

In this appendix, we list the frequently used BPDF models accordingto the order of timewhen themodel was proposed. In all these models,Fp(N,γ) is the polarized fraction of Fresnel reflectance (see Appendix A),θs and θv are the zenith angle of solar and view, and μs= cos(θs), μv=cos(θv).

B.1. Vegetation physical model (1991)

A simple physical model (Rondeaux and Herman, 1991) was pre-sented upon the hypothesis that leaves secularly reflect light accordingto the Fresnel equations which became the core part of the othermodels.

Rvegp ¼ Fp N;γð Þ

4 μs þ μvð Þ ðB1Þ

B.2. Bare soil physical model (1995)

Unlike the vegetation, the attenuation on the incident and outgoingpath of bare soil surface could be ignored. A model (Bréon et al., 1995)for bare soil was presented assuming that the ground composed of iso-topically distributed facets which can be simulated by hemispheres ofvarying radii.

Rsoilp ¼ Fp N;γð Þ

4μs:μvðB2Þ

B.3. Nadal-Bréon model (1999)

A semi-experimental BPDFmodel (Nadal and Bréon, 1999) was pro-posed based on the data retrieved by POLDER from 1996 to 1997,whichwas also used in the processing of aerosol product for POLDER.

Rp ¼ α 1− exp −βFp N;γð Þμs þ μv

� �� �ðB3Þ

where α and β are two parameter.

B.4. Waquet model (2009)

Based on the statistic of huge data, Nadal-Bréon model did not con-sider the shadowing effect directly. One model (Waquet et al., 2009b)including a shadowing function and scale parameter was developed.

Rp ¼ ξ� Fp N;γð Þ � S θsð Þ � S θvð Þ ðB4Þ

S θð Þ ¼ 2

1þ erf ρð Þ þ ρffiffiffiπ

p� �−1 exp −ρ2ð ÞðB5Þ

ρ ¼ σffiffiffi2

p� �‐1cot θð Þ ðB6Þ

where S(θ) is the shadowing function to describe the attenuation of po-larized reflectance in different view directions,ξ is the scale parameterand the roughness of surface can be described byσ.The definition oferror function (erf) is

erf xð Þ ¼ 1ffiffiffiπ

p ∫x−xe−t2dt ¼ 2ffiffiffi

πp ∫x0e

−t2dt ðB7Þ

B.5. Maignan model (2009)

Maignan et al. (2009) proposed a BPDFmodel with only one param-eter and evaluate the accuracy using the data of different cover types

from PARASOL BRDF except the urban type. Considering the fact thatthe polarized reflectance decreases with the NDVI, this model inserstthe NDVI into the function.

Rp ¼ C exp − tanαIð Þ exp −vð ÞFp N;γð Þ4 μs þ μvð Þ ðB8Þ

where v is the NDVI, αIis the incidence angel. The presence of verticalprotuberances on the leaf surface was accounted for by the followingfunction:

K αI ; kð Þ ¼ exp −k tan αIð Þð Þ ðB9Þ

To make the entire model linear, the above function was replacedwith a linear function (Maignan et al., 2009)

A exp − tan αIð Þð Þ ðB10Þ

B.6. Litvinov model (2011)

Litvinov et al. (2011) presented a three-parameter BPDF model andused the RSP data to analyze the error (Litvinov et al., 2011). This modeltook into account the influence of NDVI and surface slope distribution.This model adapted one simple experimental function to describe theshadowing effect.

Rp ¼ απFp N;γð Þ4μn μs þ μvð Þ f nv;nsð Þ f sh γð Þ ðB11Þ

f nv;nsð Þ ¼ 1πμ3

n2σ2 exp −1−μ2

n

μ2n2σ2

� �ðB12Þ

f sh γð Þ ¼ 1þ coskγ π−γð Þ2

� �3

ðB13Þ

μn ¼ nzv þ nz

s

jnv þ nsj ðB14Þ

ns ¼ sinθs cosφs; sinθs sinφs; cosθsð Þ ðB15Þ

nv ¼ sinθv cosφv; sinθv sinφv; cosθvð Þ ðB16Þ

where f(nv,ns) is the Gaussian function to describe the distribution offacets over slope orientation, fsh(γ)is the shadowing function with afree parameter kγ(0bkγb1) controlling the width of shadowing region.

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