Error Analysis for a Temperature and Emissivity Retrieval Algorithm for Hyperspectral Imaging Data Christoph Borel, PhD [email_address] , http://cborel.net ARTEMISS The 2nd I nternat i onal Sympos i um on Recent Advances i n Quant i tat i ve Remote Sens i ng: RAQRS ' II
Multi-spectral temperature-emissivity separation
Hyper-spectral temperature-emissivity separation
In-scene atmospheric correction
Spectrally smooth emissivity retrieval algorithm
A method to find the best atmosphere
Error analysis of the ARTEMISS algorithm
Retrieving spectral smile using spectral angle mapping analysis
ARTEMISS A utomatic R etrieval of T emperature and EMI ssivity using S pectral S moothness ( ARTEMISS )
Measured radiance in the thermal infrared L down L ground L path T ground , ε Problem: What is the emissivity and temperature?
Underdetermined problem ( Realmuto, 1990 ):
Given are N spectral measurements – determine N+1 unknowns (N emissivities and one temperature)
Multi-spectral temperature-emissivity separation
Assumed channel 6 emittance model: Kahle et al., 1980;
thermal log and alpha residual: Hook et al., 1992 , and
Mean-Maximum Difference (MMD): Matsunaga, 1993
Hyper-spectral temperature-emissivity separation
In-Scene Atmospheric Correction (ISAC): Johnson and Young, 1998 and 2002
Autonomous Atmospheric Compensation by Gu et al, 1999
In-scene atmospheric correction (ISAC) Radiance a blackbody would have at λ : B(λ,T B ) Measured radiance: L m (λ) Intercept ~ L p (λ) Graybody pixels ( ε <1) Blackbody pixels ( ε =1) Slope ~ ((λ) Measured radiance: Scatterplot determines transmission and path radiance: T B (i,j)=B -1 (λ 0 ,Lm(λ 0 ,i,j))/ε0
ISAC algorithm steps in detail
Steps for each pixel (i,j) do:
Select a wavelength λ 0 such that the transmission through the atmosphere is high ( ~1 ) and the path radiance is negligible (L p ~0).
Compute the apparent brightness temperature: T B (i,j)=B -1 (λ 0 ,L m (λ 0 ,i,j))/ε 0 (assume ε 0 =0.95 )
Create scatterplot with: x=B(λ,T B (i,j)) and y=L m (λ,i,j)=B(λ,T B (i,j)) +L p
Fit a straight line to the upper boundary of the points (next view graph has details)
Slope is proportional to transmission (λ)
Intercept (B(λ,0)=0 proportional to the path radiance L p . The emissivity retrieved by the ISAC method is then given by: ε(λ,i,j)=[L m (λ,i,j)-L p (λ)]/[B(λ,T B (i,j)) (λ)].
Transmission is unity and the path radiance is zero for λ= λ 0 .
For wavelengths where the transmission is higher than at λ 0 , estimated transmission will rise above unity. Negative path radiances possible too.
Schemes ( Johnson and Young, 1998 ) exist to iteratively fix the transmission and path radiance to make them physically realistic.
Fitting line to upper points in scatterplot
Steps for each wavelength λ do:
Fit a linear regression to points (x,y)=( B(λ,T B (i,j)), L m (λ,i,j))
Discard the points below the fit: y fit (x)=a*x+b
Repeat steps a&b for the points above the fit only until a fraction of points are left. The coefficient a is proportional to the transmission and the intercept b is proportional to the path radiance L p .
Iter=1 Iter=2 Iter=3 Iter=4
Example of ISAC retrieved transmssion and path radiance using simulated data ISAC transmission fits well to “true” transmission ISAC path radiance has offset and scaling errors
Atmospheric transmission and path radiance Note: The atmospheric features have sharp absorption features compared to emissivities. Modtran 4 computed τ and L p for variable water vapor amount and temperature profiles. Example of tropical atmosphere
CO2 mixing ratio,
Model atmosphere (TROPICAL, MLS, MLW, SAS, SAW, USS)
Temperature profile offset
RH scaling factor
Original smooth emissivity retrieval algorithm (Borel, 1997) Find a temperature T opt and so that the variance σ is minimized:
New version of TES based on minimizing measured and modeled sensor radiances (Borel, 2003)
Compute emissivity ε :
Where estimated ground temperature T est :
Compute the RMSE σ of the measured and simulated radiance or:
Where is the 3-point smoothed emissivity
Summary of useful error terms
Method to find the best atmosphere from ISAC
Perform spectral matching:
Spectral angle mapper (SAM):
Linear regression analysis – find maximum R 2
Select the x% best atmospheres as candidates
Size of symbols ~ SAM or ~ R 2 Best fitting atmospheres differ by Ozone amount
A utomatic R etrieval of T emperature and EMI ssivity using S pectral S moothness ( ARTEMISS ) algorithm Minimize:
Simple sensitivity study
Effect of band center shifts on radiance errors The RMS radiance error for a soil at 285 º K observed from space under different columnar water vapor amounts ranging from 1.14 to 7.41 g/cm 2 as a function of spectral shifts in channel spacings.
Full-Width-Half-Maximum effect on radiance error The RMS radiance error for a soil at 285 º K observed from space under different columnar water vapor amounts ranging from 1.14 to 7.41 g/cm 2 as a function of FWHM scaling factor .
Sensitivity study for ARTEMISS
Effect of spectral shifts on T and ε The RMSE and mean temperature retrieval error(left) and the RMSE and mean emissivity retrieval error as a function of spectral shift. The mean temperature error increases to over 1 º K for spectral shifts as small as 1/20th of a channel spacing. Wrong atmosphere causes temperature offset
Effect of Noise on Temperature T and Emissivity ε Example of the growth of the RMS temperature and emissivity error as a function of sensor noise.
RETRIEVING SPECTRAL SMILE USING SPECTRAL ANGLE MAPPING ANALYSIS
Spectral response of hyperspectral sensors can change – how can we determine spectral shifts and FWHM changes from the data itself?
Use transmission τ ISAC estimate from ISAC and compare to LUT.
Break up spectral range into K intervals: τ ISAC,k
Compute MODTRAN transmission convolved with a sensor response function for N different spectral shifts on the waveband centers and M FWHM multipliers
Normalize the K x M x N base vectors S k,m,n
Compute spectral angle SAM k,m,n between τ ISAC,k and S k,m,n for all k, m and n
3-D volume visualization of SAM k,m,n
The spectral angle mapper (SAM) is defined as:
The brightness is proportional to - log[SAM(S k,m,n , τ k )].
The X axis is the spectral interval (K), the Y axis is the spectral offset (N) and the Z axis is the FWHM multiplier (M).
Simulated retrieval of a spectral smile and FWHM variation with band number
Obtain an estimate of the FWHM multiplier variation as a function of the spectral interval by identifying for each k and n which FWHM multiplier had the best match.
Fit a line or polynomial with selectable order to the smallest SAM values to find the optimum FWHM multiplier.
Interpolate the 3-D SAM cube using the FWHM multipliers at each spectral interval index k and spectral shift index n .
Fit a line or polynomial of selectable order to the smallest SAM values to find the optimum spectral smile.
ARTEMISS algorithm uses as the main criterion the RMSE between the measured and simulated at sensor radiance where the emissivity has been smoothed to retrieve temperature and emissivity.
More than 128 channels are needed to retrieve temperature, emissivities, and atmospheric parameters.
A good atmospheric correction is a necessary condition to retrieve accurate surface temperatures and emissivities.
The spectral calibration accuracy is crucial to retrieve reasonable temperatures and emissivities.
Developed a spectral calibration method which is able to retrieve spectral shifts and FWHM of sensors with more than 128 bands to the required accuracy.
Our thanks go to Dr. Ronald Lockwood and Dr. Michael Hoke from the Air Force Research Laboratory , Hanscom AFB, MA, which supported this research during the author’s year as a distinguished AFRL National Laboratory Fellow and later under BAA contracts F19628-03-0066 and FA8718-05-C-0065.
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