## Abstract

This work analyzed and addressed the estimate of the broadband emissivities for the spectral domains 3-14μm (${\epsilon}_{3-14}$) and 3-∞μm (${\epsilon}_{3-\infty}$). Two linear narrow-to-broadband conversion models were proposed to estimate broadband emissivities ${\epsilon}_{3-14}$ and ${\epsilon}_{3-\infty}$ using the Moderate Resolution Imaging Spectroradiometer (MODIS) derived emissivities in three thermal infrared channels 29 (8.4-8.7μm), 31 (10.78-11.28μm) and 32 (11.77-12.27μm). Two independent spectral libraries, the Advanced Spaceborne Thermal Emission Reflection Radiometer (ASTER) spectral library and the MODIS UCSB (University of California, Santa Barbara) emissivity library, were used to calibrate and validate the proposed models. Comparisons of the estimated broadband emissivities using the proposed models and the calculated values from the spectral libraries, showed that the proposed method of estimation of broadband emissivity has potential accuracy and the Root Mean Square Error (RMSE) between estimated and calculated broadband emissivities is less than 0.01 for both ${\epsilon}_{3-14}$and ${\epsilon}_{3-\infty}$.

© 2011 OSA

## 1. Introduction

Surface broadband emissivity is an essential parameter for estimating the surface upward longwave radiation, which is an important component of the surface radiation budget and also an important parameter for numerical weather predictions and hydrological models [1]. Knowledge of the surface broadband emissivity is therefore valuable. Mostly, the broadband emissivity may vary significantly since the spectral emissivity variation ranges from 0.7 to 1.0 for bare soils and rocks in the 8-12μm spectral domain, while there is smaller variation in the 5-8μm and 12-14μm domains.

Recently spaceborne thermal infrared multispectral sensors, such as the Moderate Resolution Imaging Spectroradiometer (MODIS), the Advanced Very High Resolution Radiometer (AVHRR) and the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER), allow the estimation of spectral emissivities from local to global scales. There have been a lot of researches carried out on estimating the broadband emissivity from those sensors with narrowband satellite measurements [2–5]. Many efforts, however, focused on the estimate of broadband emissivity for some specified spectral domains because of the limitation of the measured spectral emissivity in the 3.0-14μm region.

This paper will be devoted to estimate the surface broadband emissivities in the spectral regions 3-14μm and 3-*∞*μm, respectively, from the measurements of MODIS Thermal Infrared (TIR) channels. Section 2 describes the method to retrieve broadband emissivity in TIR channel. Section 3 presents some preliminary results. Finally, conclusions are given in the last section.

## 2. Method

#### 2.1 Basic theory

On the basis of the Stefan-Boltzmann law, for natural objects, the total emitted radiant flux *ϕ* can be obtained by integrating the Planck function over the whole electromagnetic spectrum,

*θ*is the viewing zenith angle,

*φ*is the azimuth angle, ${B}_{\lambda}(T)$ is the Planck radiance function for a blackbody at temperature

*T*and ${T}_{\lambda}(\theta ,\phi )$ is the directional radiometric surface temperature depending on the wavelength

*λ*and the angle of observation

*θ*and

*φ*,

*Ω*is the solid angle of the whole upward hemisphere with $d\Omega =\mathrm{sin}\theta d\theta d\phi $, ${\epsilon}_{w}$ is the broadband emissivity in the whole spectral region and

*σ*is the Stefan-Boltzmann’s constant (5.67 × 10

^{−8}W m

^{−2}K

^{−4}).

On the other hand, the emissivity ${\epsilon}_{\lambda}$ is defined as the ratio of the emitted radiance ${E}_{\lambda}$ at wavelength *λ* and the blackbody emission ${B}_{\lambda}(T)$ at wavelength and temperature*T*.

Combining Eqs. (1) and (2), we get the definition of broadband emissivity written by

The channel emissivity ${\epsilon}_{i}$ for a specified sensor is consequently defined as [6]

*i*. Combining Eqs. (3) and (4) and assuming the spectral response is rectangle, the broadband emissivity can also be written by

_{i}nearly independent on the surface temperature.

#### 2.2 Algorithm development for MODIS data

As there are no available broadband emissivity measurements in coincidence with channel emissivities measurements, the only possible way to develop the linear relationship of Eq. (5) is to use numerical simulation with spectral library database measured in laboratory so far. The Advanced Spaceborne Thermal Emission Reflection Radiometer (ASTER) Spectral Library (http://speclib.jpl.nasa.gov/) including contributions from the Jet Propulsion Laboratory (JPL), Johns Hopkins University (JHU) and the United States Geological Survey (USGS) is used to simulate the broadband emissivity and MODIS channel emissivities. The library includes spectra of rocks, minerals, lunar soils, terrestrial soils, manmade materials, meteorites, vegetation, snow and ice covering the visible through thermal infrared wavelength region (0.4-14.0 μm) [7]. The spectra of rocks, soils, vegetation, water, snow and ice, which are essential components of the terrestrial ecosystem, are collected in this study. In total 268 spectral samples from JHU, JPL and USGS libraries are used to develop our algorithm. The spectra of the fine powdered samples of rock in JHU were not included in this study because they exist rarely in the natural surfaces. The ASTER library contains directional hemispherical spectral reflectance ${\rho}_{\lambda}$, we therefore converted to spectral emissivity ${\epsilon}_{\lambda}$ using Kirchhoff’s law, ${\epsilon}_{\lambda}=1-{\rho}_{\lambda}$.

The general properties of ASTER spectra are displayed in Fig. 1 . Figure 1a shows the spectral variation of the average emissivity calculated from the ASTER spectral data set and Fig. 1b displays the corresponding spectral variation of the standard deviation of this data set for different types of surfaces.

It is interesting, from Fig. 1, to note the following for a given type of surface: (1) the variation range is larger both over 3-5μm and 8-10μm regions than that of in other wavelength regions. (2) For a given wavelength, the larger the variation of emissivity is, the smaller the average emissivity is. (3) The longer the wavelength is, the smaller the spectral variation of emissivity is.

The objective of the present work is to estimate the broadband emissivity for the whole electromagnetic spectrum. Wang et al. [4] showed that the use of the narrowband emissivity in one channel instead of the broadband emissivity may result in large errors (up to 100 W/m^{2}) on the calculated surface longwave radiation. On the other hand, there are few spectral measurements available beyond 14.0 μm for natural objects because of the limitation of the measured instruments, the strong atmospheric absorption in the longwave TIR region. In addition, there are also very narrow spectral channels in the atmospheric windows for satellite sensors. The issues are: (1) which spectral domain’s emissivity can best represent the broadband emissivity in the whole electromagnetic for the calculation of the Earth emitted radiant flux without losing any accuracy? (2) how to estimate the broadband emissivity with very limited narrowband emissivities?

In order to answer the first question mentioned above, we split the electromagnetic spectra into several spectral domains and compute the contribution of each spectral domain to the overall value of the Earth emitted radiant flux. Figure 2 shows the fraction of the Earth emitted radiant flux in different spectral intervals for a blackbody with temperature ranging from 260K to 340K. From this figure we can see that the contribution of the window spectral region (8-14mm) is less than 40%, and the spectral interval 14-30μm to the overall flux is very significantly, which is the main component at temperature lower than 300K. Meanwhile, the contributions of the spectral intervals 30-50μm and 50-100μm are also important and shall not be ignored in the estimation of the surface longwave radiation.

In addition, Table 1 gives the results of the contribution of each spectral interval to the emitted flux for a blackbody at 300K. It is worthy to note that the spectral domains in which we have spectral measurements of the emissivity (3-14μm) account only for 51.5% of the emitted radiant flux while the unknown spectrum accounts for 48.4% of this flux. The contribution of the spectral interval 3-100μm to the emitted flux is more than 99%. So it is concluded that to minimize the impact of the emissivity signature in the radiant flux determination, the broadband emissivity with integrated spectral range should be taken into account to reach to 100μm. This figure also describes the weight of each spectral interval to the whole electromagnetic spectrum, which will be taken into account in the development of the linear parameterization of narrow-to-broadband emissivity conversion in Eq. (5).

On the basis of the above analysis, we calculate the broadband emissivities ${\epsilon}_{3-14}$ for spectral interval 3-14μm using Eq. (3) from ASTER spectral library for all the samples described above, respectively. We assume that the temperature T = 300K in the calculation, because the difference of broadband emissivity caused by the variation of target’s temperature over the range from 260K to 340K is smaller than 0.005 in our computation. For the calculation of broadband emissivity for 3-*∞*μm, we assume that there is no spectral variation of the emissivity in the wavelength larger than 14μm. This approximation is based on the analysis of the actual spectra depicted in Fig. 1. It has been observed that the spectral variation is smaller and smaller when the wavelength is larger and larger. We take therefore a mean value of the spectral emissivities in the spectral domain 12.5-13.5μm as the spectral emissivity for wavelength larger than 14μm for each sample.

On the other hand, we simulated MODIS channel emissivities of band 29 (8.4-8.7μm), 31 (10.78-11.28μm), and 32 (11.77-12.27μm) using Eq. (4) from ASTER spectral library for all the samples of rocks, soils, vegetation, water, snow and ice. We then got three emissivity data sets. One is broadband emissivity for spectral domain 3-14μm, one is broadband emissivity for spectral domain 3-*∞*μm, and another one is narrowband emissivities of MODIS channel 29, 31, and 32. As predicted by Eq. (5), the broadband emissivity is a linear combination of the MODIS channel emissivities. The statistical regression used here is based on the Enter regression method in the Statistical Package for the Social Sciences (SPSS) procedure. The channels are added as features to a multi-linear model. The results of the regressions are given as

*i*, and ${a}_{i}$, ${b}_{i}$ (i = 0, 3) are the statistical regression coefficients, which are listed in Table 2 .

## 3. Results and discussion

Figure 3 shows the comparisons of the broadband emissivities calculated by Eq. (3) with those estimated through Eq. (7) for ${\epsilon}_{3-14}$ and Eq. (8) for ${\epsilon}_{3-\infty}$ from ASTER spectral library. The error is large in some kinds of rocks, especially in the samples of igneous rocks, such as unaltered volcanic tuff with maximum error of −0.025 for ${\epsilon}_{3-\infty}$ and anthophyllite mica schist with maximum error of 0.020 for ${\epsilon}_{3-14}$. Good agreements are observed between estimated broadband emissivities (use of Eqs. (7) and 8) and calculated ones (use of Eq. (3) for JHU and JPL soils with most differences smaller than 0.01. There exists a little underestimation of ${\epsilon}_{3-14}$ for samples with higher emissivity (around 0.99), such as vegetation, water and snow/ice. This may be caused by the fact that there exist very small numbers of spectra for these types used in the regression. There are only four vegetation samples including conifers, deciduous, dry grass and green grass, four water samples including distilled water, sea foam, sea water and tap water, and three snow samples including coarse, medium, and fine granular snow and one sample of ice in JHU library. But from Fig. 3 we can see that the Root Mean Square Error (RMSE) between estimated and calculated broadband emissivities is 0.005 for both ${\epsilon}_{3-14}$ and ${\epsilon}_{3-\infty}$, with all the 268 samples used.

To test and validate the narrow-to-broadband emissivity conversions (Eqs. (7) and 8) proposed in this study, the MODIS UCSB (University of California, Santa Barbara) emissivity library of the MODIS LST group was used (http://www.icess.ucsb.edu/modis/EMIS/html/em.html). We selected 107 spectra collected by Snyder et al. [8], including 68 soil types, 28 vegetation types, 6 water types, and 5 snow/ice types. We first calculated the broadband emissivities ${\epsilon}_{3-14}$ and ${\epsilon}_{3-\infty}$ with Eq. (3), and MODIS narrowband channel emissivities 29, 31 and 32 with Eq. (4). Then we estimated the broadband emissivities ${\epsilon}_{3-14}$ and ${\epsilon}_{3-\infty}$ with our proposed models (Eqs. (7) and 8). Figure 4 shows the comparisons of the calculated broadband emissivities with the estimated values for ${\epsilon}_{3-14}$ and ${\epsilon}_{3-\infty}$, respectively.

The RMSE between calculated and estimated broadband emissivities is 0.006 for ${\epsilon}_{3-14}$ and is 0.005 for ${\epsilon}_{3-\infty}$. The maximum error is 0.029 in ${\epsilon}_{3-\infty}$ for smooth ice and the maximum error is 0.020 in ${\epsilon}_{3-14}$ for one of the playa soils. There exists an underestimation of broadband emissivity ${\epsilon}_{3-14}$ for snow/ice samples, as shown also with ASTER library data. Table 3 gives the statistical ranges of emissivity variation, the RMSE and maximum errors in both calibration and validation data sets (ASTER spectral data set for calibration and UCSB spectral data set for validation).

## 4. Conclusions

In this work, the broadband emissivities for spectral domains 3-14μm and 3-∞μm have been estimated using linear regression functions with three MODIS thermal infrared channel emissivities, respectively. Based on the analysis of the fraction of the Earth emitted radiant flux in different spectral intervals for a blackbody with temperature ranging from 260K to 340K, the broadband emissivity with integrated spectral range for the whole electromagnetic has been suggested to take into account for the determination of the Earth emitted flux.

The spectra of essential components of the terrestrial ecosystem including rocks, soils, vegetation, water, snow and ice, in the ASTER spectral library, have been used to develop the narrow-to-broadband emissivity conversion models. The results showed that the estimation of broadband emissivity is potentially accurate and the Root Mean Square Error (RMSE) between estimated and calculated broadband emissivities is 0.005 for both ${\epsilon}_{3-14}$ and ${\epsilon}_{3-\infty}$ with all the 268 samples used.

In order to test and validate the estimated accuracy of the proposed method, an independent MODIS UCSB emissivity library with 107 samples has been used. The results showed that the RMSE between estimated broadband emissivities and calculated ones is 0.006 for ${\epsilon}_{3-14}$ and is 0.005 for . ${\epsilon}_{3-\infty}$.

## Acknowledgments

This work was jointly supported by National High Technology Research and Development Program of China (2009AA122102 and 2008AA121805), and National Natural Science Foundation of China under Grant No 40801140. The ASTER Spectral Library was provided by the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, and the UCSB emissivity library was provided by the MODIS LST group. The authors would like to thank the anonymous reviewers for their valuable comments.

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