Bi-Directional Reflectance Distribution Function: BRDF Effect on Un-mixing, Category Decomposition of the Mixed Pixel (MIXEL) of Remote Sensing Satellite Imagery Data

Method for unmixing, category decomposition of the mixed pixel (MIXEL) of remote sensing satellite imagery data taking into account the effect due to Bi-Directional Reflectance Distribution Function: BRDF is proposed. Although there is not so small BRDF effect on estimation mixing ratios, conventional unmixing methods do not take into account the effect. Through experiments, the effect is clarified. Also the proposed unmixing method with consideration of BRDF effect is validated.


INTRODUCTION
The pixels in earth observed images which are acquired with Visible to Near Infrared: VNIR sensors onboard remote sensing satellites are, essentially mixed pixels (mixels) which consists of several ground cover materials [1].Some mixel model is required for analysis such as un-mixing of the mixel in concern [2], [3].Typical mixel is linear mixing model which is represented by linear combination of several ground cover materials with mixing ratio for each material [4].It is not always true that the linear mixel model is appropriate [5].Due to the influences from multiple reflections between the atmosphere and ground, multiple scattering in the atmosphere on the observed radiance from the ground surface, pixel mixture model is essentially non-linear rather than linear.These influences are interpreted as adjacency effect [6], [7].
Although there is not so small BRDF effect on estimation mixing ratios, conventional unmixing methods do not take into account the effect.Method for unmixing, category decomposition of the mixed pixel (MIXEL) of remote sensing satellite imagery data taking into account the effect due to Bi-Directional Reflectance Distribution Function: BRDF is proposed.In order to take into account BRDF effect, Minneart Reflectance Model: MRM is utilized for representation of BRDF.Through experiments, the effect is clarified.Also the proposed unmixing method with consideration of BRDF effect is validated.
The following section, the proposed method is described followed by the experiments.Then conclusion is described together with some discussions.

A. Surface Reflectance Models
BRDF is defined in equation (1).
Lambertian surface, on the other hand, is defined in equation ( 2). ( Where I n denote incident irradiance in the normal direction while I θ denotes reflected radiance in direction of θ.Therefore, the Lambertian surface reflectance is constant at all direction, for entire hemisphere. Minnerart reflection is defined in equation ( 3).

B. Unmixing, Category Decomposition Method
One single pixel of remote sensing satellite imagery data, P can be represented as combination of weighted spectral characteristics, H j of the considerable ground cover targets included in the Instantaneous Field of View: IFOV with their mixing ratios, a j as is shown in equation ( 4).

C. Monte Carlo Ray Tracing Simulation
In order to show a validity of the proposed non-linear mixel model, MCRT simulation study and field experimental study is conducted.MCRT allows simulation of polarization characteristics of sea surface with designated parameters of the atmospheric conditions and sea surface and sea water conditions.Illustrative view of MCRT is shown in Figure 1.Photon from the sun is input from the top of the atmosphere (the top of the simulation cell).Travel length of the photon is calculated with optical depth of the atmospheric molecule and that of aerosol.There are two components in the atmosphere; molecule and aerosol particles while three are also two components, water and particles; suspended solid and phytoplankton in the ocean.
When the photon meets molecule or aerosol (the meeting probability with molecule and aerosol depends on their optical depth), then the photon scattered in accordance with scattering properties of molecule and aerosol.The scattering property is called as phase function 1 .In the visible to near infrared wavelength region, the scattering by molecule is followed by Rayleigh scattering law [8] while that by aerosol is followed by Mie scattering law [8].Example of phase function of Mie scattering is shown in Figure 2    For simplifying the calculations of the atmospheric influences, it is assumed that the atmosphere containing only molecules and aerosols.As shown in Figure 3, this assumption is not so bad.Thus the travel length of the photon at once, L is expressed with equation ( 8).
Where Z max, τ, RND(i) are maximum length, altitude of the atmosphere, optical depth, and i-th random number, respectively.In this equation, τ is optical depth of molecule or aerosol.The photon meets molecule when the random number is greater than τ.Meanwhile, if the random number is less than τ, then the photon meats aerosol.The photon is scattered at the molecule or aerosol to the direction which is determined with the aforementioned phase function and with the rest of the travel length of the photon.

A. Preliminary Simulation Studies
A mixed pixel model which consists of two categories is created.Also two different surface reflectance models, Minneart and Lambertian surface models are assumed as shown in Figure 4. Alunite and Cheat grass are selected for the categories.These spectral characteristics are given by USGS spectral library.Mixel with 50% of Alunite and 50% of Cheat grass is created with the surface slope of 20 degree.Observed radiance from the mixel is derived from MCRT at wavelength of 550 and 650 nm.Unmixing is attempted with Minneart and Lambertian surface models.Spectral characteristics of Alunite and Cheat grass are shown in Figure 5 (a) and (b).Also angle distribution of Minneart surface with Minneart coefficients of 0.3, 0.8 and 1.0 (Lambertian surface) is shown in Figure 6.Due to the fact that the surface slope angle is set at 20 degree, reflectance between Lambertian and Minneart reflection models show no difference at 20 and 110 degree of observation angles.Also Figure 7 shows reflectance as a function of slope.Therefore, 2.25% of estimation error is observed for the mixing ratio for the Lambertian surface model.The results show that there is significant error on mixing ratio estimation when BRDF is not taken into account.

B. Simulation Studies
Visible to Near Infrared: VNIR radiometer onboard remote sensing satellite is simulated with MCRT.VNIR consists of three bands, 400, 700, and 900nm with IFOV of 15m.Observation angle is set at zero zenith angle (nadir view) while solar zenith angle is set at 20 degree.
Ground surface is assumed to be covered with soil) default materials of MODTRAN.Within a IFOV, there are three soils with the different reflectance, 0.1, 0.3, and 0.5 of Lambertian and MInneart surfaces.Minneart coefficients are 1.0 for Lambertian surface and 0.8 for Minneart surface, respectively.
Mid. Latitude Summer of the atmospheric model is selected from MODTRAN.Optical depth of atmospheric molecule is set at 0.0003 while that of aerosol particle is set at 0.2 at 700 nm.Refractive index of aerosol particle is set at 1.44i 0.005.Junge size distribution of aerosol particle is assumed with Junge parameter of 3.0.Then phase function (scattering characteristic) can be derived from Mie code which is included in MODTRAN.Top of the Atmosphere: TOA radiance, then calculated for pixel by pixel basis.Table 1 shows the TOA radiance in unit of mW/cm 2 /str.Also the TOA radiance of the assumed mixels with the different mixing ratio is calculated.Mixing ratio of 0.1, 0.3, and 0.5 of reflectance soils is 0.5, 0.3, and 0.2, The results are 0.295 (@400nm), 0.222 (@700nm), and 0.185 (@900nm), respectively for Lambertian surface while those for Mineart surface are 0.262 (@400nm), 0.201 (@700nm), and 0.175 (@900nm), respectively.
Using the aforementioned TOA radiance of the assumed mixels for both Lambertian and Minneart surfaces, mixing ratio of the different reflectance of soils are estimated based on the proposed unmixing method of category decomposition.The result is as follows, 6.65% (soil #1), 82.74% (soil #2), 10.61% (soil #3) for Lambertian surface while 49.95% (soil #1), 27.59% (soil #2), 22.46% (soil #3) for Minneart surface.This implies that the estimated mixing ratios are appropriate when BRDF is taken into account (Minneart surface) while the estimated mixing ratios have significant errors if BRDF is not taken into account (Lambertian surface).

IV. CONCLUSION
Method for unmixing, category decomposition of the mixed pixel (MIXEL) of remote sensing satellite imagery data taking into account the effect due to Bi-Directional Reflectance Distribution Function: BRDF is proposed.Although there is not so small BRDF effect on estimation mixing ratios, conventional unmixing methods do not take into account the effect.Through experiments, the effect is clarified.Also the proposed unmixing method with consideration of BRDF effect is validated.
The estimated mixing ratios are appropriate when BRDF is taken into account (Minneart surface) while the estimated mixing ratios have significant errors if BRDF is not taken into account (Lambertian surface).
angle and observation angle, respectively while k denotes Minneart coefficient.If k=1, it is totally equation to Lambertian surface.

Fig. 1 .
Fig. 1.Illustrative view of MCRT for the atmosphere and sea water (a) while that of Rayleigh scattering is shown in Figure 2 (b).

Fig. 2 .
Fig. 2. Phase functions for Mie and Rayleigh scattering In the atmosphere, there are absorption due to water vapor, ozone and aerosols together with scattering due to the atmospheric molecules, aerosols.Atmospheric Optical Depth: AOD (optical thickness) in total, Optical Depth: OD due to water vapor (H 2 O), ozone (O 3 ), molecules (MOL), aerosols (AER), and real observed OD (OBS) are plotted in Figure 3 as an example.

Fig. 3 .
Fig. 3. Example of observed atmospheric optical depth in total and the best fit curves of optical depth due to water vapor, ozone, molecules, and aerosols calculated with MODTRAN of atmospheric radiative transfer software code..

Figure 7 (
a) shows the calculated reflectance of Alunite as a function of slope angle with the different parameters of Minneart coefficients, 0.3, 0.8, and 1.0 while Figure 7 (b) shows that of Cheat grass as a function of slope angle with the different parameters of Minneart coefficients, 0.3, 0.8, and 1.0 at the wavelength of 550 nm.Therefore, slope effect is quite dependent on Minneart coefficients.Unmixing results based on Minneart reflectance model show correct mixing ratio of 50 versus 50% while those based on Lambert reflectance model show 52.25 versus 47.75%.