Extraction of incident irradiance from LWIR hyperspectral imagery Pierre Lahaie, DRDC Valcartier 2459 De la Bravoure Road, Quebec, Qc, Canada

ABSTRACT The atmospheric correction of thermal hyperspectral imagery can be separated in two distinct processes: Atmospheric Compensation (AC) and Temperature and separation (TES). TES requires for input at each pixel, the ground leaving and the atmospheric downwelling irradiance, which are the outputs of the AC process. The extraction from imagery of the downwelling irradiance requires assumptions about some of the pixels’ nature, the sensor and the atmosphere. Another difficulty is that, often the sensor’s spectral response is not well characterized. To deal with this unknown, we defined a spectral mean operator that is used to filter the ground leaving radiance and a computation of the downwelling irradiance from MODTRAN. A user will select a number of pixels in the image for which the emissivity is assumed to be known. The emissivity of these pixels is assumed to be smooth and that the only spectrally fast varying variable in the downwelling irradiance. Using these assumptions we built an algorithm to estimate the downwelling irradiance. The algorithm is used on all the selected pixels. The estimated irradiance is the average on the spectral channels of the resulting computation. The algorithm performs well in simulation and results are shown for errors in the assumed emissivity and for errors in the atmospheric profiles. The sensor noise influences mainly the required number of pixels.

Keywords: Hyperspectral imagery, atmospheric correction, temperature emissivity separation


The atmospheric correction of thermal hyperspectral imagery aims at extracting the temperature and the emissivity of the material imaged by a sensor in the long wave (LWIR) spectral band. It can be divided in two parts: atmospheric compensation (AC) and temperature emissivity separation (TES). TES algorithms such as ARTEMISS [1, 2] and DEFILTE [3] require for input the ground leaving radiance and the atmospheric downwelling irradiance. These inputs are produced by atmospheric compensation of sensor measured radiance. An important difficulty is the estimation from imagery of the atmospheric downwelling irradiance. This paper proposes an approach to perform that task. The downwelling irradiance is difficult to estimate because, very often the spectral radiance emanating from materials that are smooth spectrally is very high and therefore the content of reflected radiance is small. Using spectrally smooth pixel is useful to reduce the impact of spectral features in the corrected spectra. The algorithm makes use of a known or at least an assumed spectral emissivity for a number of pixels and of an assumed atmospheric profile. The atmospheric profile is useful to provide a basis for the estimated downwelling irradiance. The impact on the TES application will be to reduce the error of the estimated temperature.

The paper is organized as follow: We first describe the algorithm; then the simulation used to assess the sensitivity of the algorithm to noise and to the error on atmospheric profile. The results of the simulations are provided and described, finally we conclude.

Equation (1) shows the sensor measured radiance and equation (2) is the ground leaving radiance.



Where is the radiance measured by the sensor, is the ground leaving radiance (The radiance emitted by the pixel’s material, composed partly of self-radiation and of the reflection of incident ), is the , is the path radiance, is the emissivity of the pixel’s material, is the blackbody radiation at the temperature T and finally L is the incident downwelling irradiance on the pixel transformed in radiance using the assumption that the material is Lambertian. In the remainder of the document we refer to the downwelling irradiance as the radiance reflected by the material. There are often many problems in the way for extracting the downwelling irradiance. The first is that since we do not know the characteristics of the image, we have to use assumptions about the nature of some pixels’ material and temperatures. We also need some prior data about the atmospheric profiles. One other difficulty is that very often the spectral response of each sensor’s pixel is not well characterized.


We assume in the algorithm that the atmospheric compensation (AC) has been performed correctly. The AC process is the removal of the atmospheric path radiance accumulated in the air on the path from the target to the sensor and of the transmittance of the atmosphere on the same path. It usually requires the estimation of these two parameters from either use of a modelling approach, MODTRAN [4] for example or by the use of an in scene technique such as ISAC [5]. The ground leaving radiance (2) is obtained from the sensor measured radiance (1) when the path radiance and the transmittance are removed.

The emissivity is the first unknown related to the pixels that are used to estimate the downwelling irradiance. Often the emissivity of the suitable pixels in the image is not known exactly. Assumptions have to be used and contained in spectral libraries can be used when the material is known or assumed. The emissivities contained in spectral libraries are generally averages valid for a specific material. Any error in the used spectral emissivity compared to the effective emissivity of the material in the pixel will map in the whole image when TES will be performed. This is the main reason why spectrally smooth emissivities shall be used. Materials having spectrally smooth characteristics generally have a high emissivity. This will have an impact on the number of pixels that shall be used since the reflected radiance, the signal of interest here, will show a smaller signal to noise ratio.

A second problem is the fact that, the whole spectral response of each pixel of the sensor is not well known and in fact is very difficult to measure accurately and for all variations of the sensor. Assumptions about the response can be used, but, they generally give worst results. This is due in the thermal infrared to the large number of spectral absorption lines. To alleviate this, a spectral average of the sensor spectrum is computed over a window that is large enough to reduce the impact of wrong registration of the spectral bands; therefore if some lines are introduced or ignored in the computation their impact will be reduced due to the large number of lines. The same window width is used to compute the downwelling irradiance from a model like MODTRAN, this considered as being the mean of the downwelling irradiance.

We define a spectral mean operator having the following general form:


Where, is the quantity that is averaged and is the spectral index (, wavenumber or ). Starting with the ground emitted radiance we obtain:


For many materials the emissivity can be assumed to have a slow variation with wavelength, so, inside the window, it can be considered as constant. If the variation is linear, the mean will be the same as the median value and therefore a linear variation is suitable. In the case of the blackbody function, it also has a slow variation and therefore as long as the window is small enough, it can be considered constant. The spectral unit or spectral domain (wavelength, frequency or wavenumber) also has an influence for the blackbody consideration since the variation is linear for a wide range of temperatures in the thermal band. The following approximation to (4) can be used:


Subtracting (5) from (2) and isolating the downwelling irradiance yield:


In (6), the sensor measurement and its spectral average are taken directly from the image. The emissivity has to be assumed or measured or obtained from another source and the spectral average of the downwelling irradiance has to be obtained from either a direct ground measurement or an atmospheric model such as MODTRAN. There are therefore many unknowns and uncontrolled parameters for which it is required to perform a sensitivity analysis. Sensor noise is another component of the problem that needs to be assessed. The following section on simulation describes how this sensitivity analysis has been performed.


In this section we study the impact of the variation of three parameters of the system and of the sensor, the emissivity, the error on atmospheric profiles and the sensor noise. There could be very important other difficulties that are not studied example of which are the sensor calibration and the spectral registration.

We assume that a good estimation of the emissivity mean has been performed and that therefore only the variation of it around the mean have impacts. If the emissivity average used in the computation is wrong, equation (6) becomes:


Where and are respectively the real average emissivity of the material and the assumed emissivity while , and are respectively the estimated, the real and the modeled downwelling irradiance. The estimated downwelling irradiance will contain in some way the difference in spectral features between the real emissivity and the assumed emissivity. If the emissivity is spectrally smooth the spectral features will not have a high importance in the result. Independently, if there is a difference between the spectral average of the real and the modeled downwelling irradiance, it will map in the offset of the estimated downwelling irradiance. In many cases however, the atmospheric profiles provided by weather authorities will be close enough to the real profiles and MODTRAN is thought to be close to about 5% from measured atmospheric optical parameters. All the simulations have been performed in the wavenumber domain with a sensor having 128 bands extending from 744 to 1232 cm-1 with steps of 4 cm-1. This choice of units is related to the natural units used by MODTRAN. The MODTRAN simulations have been performed with a 1 cm-1 resolution and its band model.

The variation in emissivity has been performed by assuming a Gaussian shape for the covariance of the spectral emissivity. Figure 1a displays the mean of the emissivity and some realization of emissivity variations using the covariance matrix shown in figure 1b. This emissivity is for the light yellowish brown clay soil file 89p1763.txt from the ASTER spectral library [6].

Two downwelling irradiances have to be used in the simulation process , the first one is used to perform the computations of in equation (2) and the second is used to compare the effects of a difference average irradiance compared to the reference one in equation (6). The irradiances are computed using MODTRAN standard atmospheric models: The mid latitude summer model and the US Standard model. These two profiles are very different, the US standard being colder and drier than the mid latitude summer profile. The two profiles are sufficiently different to provide a good comparison of their impact on the downwelling irradiance to be extracted. Figure 2 displays the differences in downwelling irradiances computed with the two profiles in details at the sensor resolution and with a spectral average of 10 bands. 1



0.97 Emissivity value 0.96


0.94 600 700 800 900 1000 1100 1200 1300 1400 -1 Wavenumber [cm ] Figure 1a: Mean emissivity extracted from the 89p1763.txt Figure 1b: Covariance matrix used to produce emissivity file from the ASTER spectral library and some variations of variations of figure 1a that emissivity. The black line is the mean emissivity while the gray lines are random realization of that emissivity using the covariance of figure 1b.

0.14 0.14

0.12 0.12 .sr] .sr] -1 -1 0.1 0.1 .cm .cm 2 2

0.08 0.08

0.06 0.06

0.04 0.04 Irradiance to ground [W/m to ground Irradiance Irradiance to ground [W/m to Irradiance ground 0.02 0.02

0 0 700 800 900 1000 1100 1200 1300 700 800 900 1000 1100 1200 1300 -1 -1 Wavenumber [cm ] Wavenumber [cm ] Figure 2a: Downwelling irradiance (in radiance units) Figure 2b: Filtered downwelling irradiance (spectral average computed for the Mid latitude summer (black) and the US eq. (3) Computed respectively for the Mid latitude summer and Standard atmospheric profiles of MODTRAN in the sensor the US Standard atmospheric profiles resolution

The third component of the simulation is the noise. The noise level is computed at 1000 cm-1 wavenumber for a blackbody radiating at 300K and the signal to noise ratio of any single run. The last parameter is the simulation size. Since two variable parameters are present in the simulation, the emissivity and the noise, this parameter is importan t to determine how many pixels of the same material are required to estimate adequately the downwelling irradiance.


Figure 3 displays the error of computation of the downwelling irradiance The differences between the results obtained with the different spectrally averaged downwelling irradiances of figure 2b computed using MODTRAN are displayed in figure 3. 0.06 0.6

0.4 0.04 .sr]

-1 0.2 0.02 .cm 2 0 0 -0.2 Relative error -0.02 -0.4 Error on Radiance [W/m Radiance on Error -0.04 -0.6

-0.06 -0.8 700 800 900 1000 1100 1200 1300 700 800 900 1000 1100 1200 1300 -1 -1 Wavenumber [cm ] Wavenumber [cm ] Figure 3a: The error on the estimation of the downwelling Figure 3b: The relative error on the estimation of the irradiance depending on the basic atmospheric profile that is downwelling irradiance depending on the atmospheric profile used to estimate the spectral average. The black curve is for the that is used to estimate the spectral average. The black curve is mid latitude summer and the gray is for the US Standard for the mid latitude summer and the gray is for the US profiles respectively Standard profiles respectively

Figure 4 shows the output of the DEFILTE algorithm for a source having a temperature of 300K and the emissivity that has been used to estimate downwelling irradiance.

0.99 0.99

0.985 0.985

0.98 0.98

0.975 0.975 Emissivity Emissivity

0.97 0.97

0.965 0.965

0.96 0.96 750 800 850 900 950 1000 1050 1100 1150 1200 1250 750 800 850 900 950 1000 1050 1100 1150 1200 1250 -1 -1 Wavenumber [cm ] Wavenumber [cm ] Figure 4a: Emissivity computation using DEFILTE for the two Figure 4b: Emissivity comparison for the random realization of downwelling irradiances. The black curve is a random the emissivity (gray curve) and the computation of the realization of the emissivity and the simulation irradiance (mid emissivity using the estimated downwelling irradiance and the latitude summer) the gray curve is computed emissivity using DEFILTE algorithm with the mid latitude summer atmospheric the US standard downwelling irradiance and the dashed curve profile is the root emissivity

In the graphs of figure 4a, the temperature obtained for the two profiles are respectively of 300K and 300.3K for the gray curve. The simulation temperature is 300K. The differences are not very important however it can be seen that the difference in the emissivity originates mainly from the difference in the behavior of the spectral average of the downwelling irradiances. In figure 4b, the general emissivity behavior is very comparable to the emissivity used to perform the simulation. Therefore, if the modeled downwelling irradiance is close to the real downwelling irradiances the results shall be very close to the true emissivity with high frequency variations that could be filt ered out if required.

The sensor noise level has two impacts in TES. The first and most important is its direct impact on the computation of the emissivities and on the temperature when the downwelling irradiance is known and the second is on the computation of the downwelling irradiance itself using some pixels. This impact is more important if the emissivity of the pixels is high because the signal (reflected downwelling irradiance) is small at that time. The following table provides the distance, between the simulated downwelling irradiance with respect to noise level and simulation size. The distance is expressed by the following expression (8) where, is the estimated downwelling irradiance in band n and is the reference downwelling irradiance, estimated without noise:


Distance N_sim = 100 200 500 1000 1500 SNR = 100 1.0753 0.9755 0.5447 0.3389 0.3241 200 0.5397 0.3409 0.2781 0.1895 0.1316 500 0.2553 0.1742 0.1019 0.0614 0.0557 1000 0.1140 0.0698 0.0592 0.0378 0.0319 1500 0.0734 0.0543 0.0303 0.0240 0.0237

0.3 0.12

0.1 0.2 0.08

0.1 0.06

0.04 0 0.02 -0.1 0 Relative Difference Relative Difference

-0.2 -0.02

-0.04 -0.3 -0.06

-0.4 -0.08 700 800 900 1000 1100 1200 1300 700 800 900 1000 1100 1200 1300 -1 -1 Wavenumber [cm ] Wavenumber [cm ] Figure 5a: SNR = 100 and simulation size = 100 Figure 5b: SNR = 100 and simulation size = 1500

-3 x 10 0.04 6

4 0.03 2

0 0.02 -2

0.01 -4


Relative Difference 0 Relative Difference -8

-10 -0.01 -12

-0.02 -14 700 800 900 1000 1100 1200 1300 700 800 900 1000 1100 1200 1300 -1 -1 Wavenumber [cm ] Wavenumber [cm ] Figure 5c: SNR = 1500 and simulation size = 100 Figure 5d: SNR = 1500 and simulation size = 1500

Figure 5: Relative difference between an estimation of the downwelling irradiance for 4 cases of simulation size and of signal to noise ratio.


In this paper, we described an algorithm to estimate the incident downwelling irradiance from the sky and sun on the ground to feed TES algorithm with the objective to extract the temperature and the emissivity of imaged material by airborne hyperspectral sensor. The algorithm makes use of a hypothetical atmospheric profile to compute the atmospheric downwelling irradiance. This parameter is spectrally averaged in a similar way compared to the sensor measured radiances and is used to estimate the downwelling irradiance in the sensor domain.

The algorithm has been tested in simulation for emissivity variations, the impact of an error on the atmospheric profile and the sensor noise. The algorithm is robust against the emissivity variation in the tested environment but a thorough testing is required by using different emissivity and especially emissivities that are more variable and lower in value. The impact of a large error generated by the use of the wrong atmospheric model does not have a large impact. The spectral features of the computed atmospheric downwelling irradiances are mapping in the estimated downwelling irradiance but it will not have a high impact when the interesting pixel material has large emissivity features that will overcome the impact of the error. The noise of the sensor can have an impact if the emissivity that is used is high, like the emissivity that we used in the simulations. By increasing the number of pixels used to estimate the downwelling irradiance the impact of noise is reduced. We stopped at 1500 pixels for that estimation, and that corresponds to approximately 40 by 40 pixels square. The number of pixels to be used in estimation can be much higher than that if large region of similar characteristics exist in an image. For example a sensor having a signal to noise ratio from 250 to 500 can be used to estimate the downwelling irradiances.

The algorithm has been tested only in simulation. It will have to be validated experimentally with real data and further improvements will be added. The impact of the estimation of the transmittance and of the path radiance estimation and removal from the images has not been evaluated. This work is devoted to the future.


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