REMOTE SENS. ENVIRON. 42:117-121 (1992)
Two-Temperature Method for Measuring Emissivity Ken Watson U.S. Geological Survey, Denver
Spectral emissivity can be uniquely determined from radiance measurements if the object can be observed at two different temperatures. The advantage of this approach is that the spectral emissivity is determined without a priori assumptions about spectral shape. Because the different temperatures are obtained by observing the scene at two times in the diurnal cycle (optimally after midday and midnight), the method assumes that emissivity is temporally invariant. This is valid for rocks and dry soils, not well established for vegetation, and not true when changes in soil moisture occur between the measurements. Accurate image registration and satisfactory signal:noise are critical factors that limit extensive use of this method. INTRODUCTION Spectral emissivity can be determined from radiance measurements of the scene at two times in the diurnal cycle, thus at two distinct temperatures. Consider the simple relation (without atmospheric terms)
R~j=eaBa(Tj), j = 1,jmax,
(1)
where Raj is the measured spectral radiance at wavelength 2, j is the jth time, ea is the spectral emissivity (assumed to be time-invariant), and Ba(Tj)
Address correspondence to Ken Watson, U.S. Geological Survey, Denver Federal Bldg., Denver, CO 80225. Received 15 July 1992.
ISSN / 92 / $0. O0 Published 1992 by Elsevier Science Publishing Company, Inc.
is the blackbody radiance at temperature Tj, given by the Planck function B~(T)= C~L-5[exp(C2/AT) -1] -1, where C1 and C2 are the radiation constants. The problem is solvable ifjm~,> 1 because, for N radiance measurements, there are N*jm~x equations in N +jm,x unknowns (N ea's andjm~ T]s).
METHOD If the radiance is convolved with an instrument response function, Eq. (1) may be written as
S~j= E~B~j, i = 1, N a n d j = 1, 2,
(2)
where S~j is the measured signal, E~ is the spectral emissivity, B~jis the blackbody radiance at temperature Tj, and subscripts i and j denote the ith channel and thejth time. Forj = 2, Eq. (2) consists of 2N equations in N + 2 unknowns, and a solution to this problem is theoretically possible when N>I. One approach to determining E~ is to ratio the measured signal at the two times:
S,2/S,~ =B,2/B,~ = lexp(C2/2'TO - 11 [exp(Cz / 2,(T2) - 1]'
i = 1, N.
(3)
Equation (3) may then be solved for the two temperatures (T1,T2) exactly if N--2 or by some form of a successive-approximation least-squares method if N > 2 . The author has experimented with several approaches and found that the most satisfactory method is to use one equation to
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Figure la. A color composite image of band ratios (4 / 3 - red, 2 / 1 - g r e e n , 6 / 5 blue) using this algorithm for an area in the Rodeo Creek NE, Nevada 1:24,000 scale quadrangle. The Carlin mine area is the large reddish anomaly in the lower center of the image.
express one of the temperatures as a function of the other and solve for the remaining temperature using a successive approximation. The temperatures can be bounded using the fact that Emin < El < 1, where Eminis some arbitrary estimate of the lowest value of E that can occur. The solution bounds on Tj can then be obtained from this range using Eqs. (1) and (2). Once the temperatures are computed then the B~j terms can be determined, and the emittances El solved using Eq. (2). A color composite image of emissivity band ratios (4 / 3-red, 2 / 1-green, 6 / 5-blue) was computed using this algorithm for an area in northern Nevada including the Carlin gold mine (Fig. 1). The color scheme was selected to yield a display somewhat similar to the display of the decorrelation stretch technique (Gillespie et al., 1986). Areas of silicification, quartzite, and disturbed
ground around the Carlin and Blue Star mines appear red, pure carbonates are bright green, impure carbonates are bluish green, and quartz latites are dark purple. Figure 2 shows a comparison between laboratory-measured and image spectral ratios. A more comprehensive analysis of the TIMS data of this area has recently been published (Watson et al., 1990).
ADVANTAGES AND DISADVANTAGES
The two-temperature method has as its primary advantage the direct computation of spectral emissivity without assumptions about the spectral shape. One immediate constraint is that the emissivity is assumed to be temporally invariant. This condition is clearly valid for rocks and dry soils,
Two-Temperature Methodfor MeasuringEmissivity 119
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but not well established for vegetation, and definitely not true when transient changes occur between the measurements: for example, changes in ground moisture or surficial materials. The method requires that data be acquired at two times in the diurnal cycle. Because solution of Eq. (3) requires temperatures that are distinct, the data should be acquired near the maximum and minimum of the diurnal temperature range, that is, after midday and after midnight. These data must be coregistered, which can be a complex task for aircraft scanner data, particularly in mountainous terrain (Gillespie, 1980; Schowengerdt, 1983; Hummer-Miller, 1990). Misregistration at the pixel level introduces high spatial frequencies into the resulting images. Removal of scanner noise prior to the registration is critical. The most significant limitation of this method is that the solution of Eq. (3) is very sensitive to the system signal:noise ratio.
1 Figure lb. A location map to identify features in Figure la. The identifying letters correspond to a) silicification, b) quartzite, c) disturbed ground around the Carlin and Blue Star mines, d) pure carbonate, e) impure carbonate, and f) quartz latite.
CAVEATS To test the sensitivity of the two-temperature approach, a radiance simulation model (Watson, in preparation) was constructed using a random number generator to vary emittance, temperature, and additive system noise. The statistical ranges of the computed radiances were chosen to match an observed data set. Large numbers of samples and repeated cases were used to insure the validity of results. The signal:noise of the radiance was compared to that of the computed temperature using the ratio of the mean to the standard deviation of the estimate. For very high signal:noise (>104 ) the temperature accuracy is about 1/200th of the radiance value. However, at lower signal:noise the temperature accuracy is asymptotic to a value of about 40. Because the nominal signal:noise of the TIMS instrument is = 500:1, the asymptotic value applies. Improving
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the system signal:noise by a factor of 10 or more is unlikely in the near future, so that this limitation is likely to continue. It was concluded from these tests that temperature accuracies of roughly 300 / 40 or 7.5 K can be achieved using the twotemperature method. This produces an emissivity accuracy of only about 12%. The high sensitivity of the two-temperature m e t h o d to system noise arises as a result of the properties of the Planck function near its maxim u m value. If the Wien approximation is used, the problem becomes more apparent. In that case the solution degenerates from two separate temperatures to a single quantity (1/T2-1TI). If residuals are computed from the exact form (with
noise present), the contours, in a simple two-axis plot of 1/T2 vs. 1/T1, are highly elliptical with the long axis along the line 1 / T2 - 1 / T] = const. As the signal:noise rises the contours become more circular, thus indicating a distinct solution for the two temperatures. It is also possible, however, to combine the two-temperature method with the emissivity-ratio method that was described elsewhere in this issue. This combined method has been successfully applied to mineral exploration in Nevada (Watson et al., 1990). The approach removes the signal: noise constraint of the two-temperature method, but retains both the invariant emissivity assumption and registration problems and restricts the
Two-Temperature Method for Measuring Emissivity
spectral information to spectral ratios. It also introduces a means to estimate those atmospheric terms that can be approximated by stationary forms. Equation (1) can be rewritten to include atmospheric terms: Raj = ra[eaBa(Tj) + (1 - e~)Hskyaj} + Hpatha~
(4)
where ra is the atmospheric transmission coefficient along the path from the ground to the detector and Hsky~j and Hpath o are respectively the flux of radiation incident on the ground from the sky and a small correction term representing radiant flux generated in the atmosphere along the path from the ground to the detector. The temporal difference in the measured radiance may be used first to estimate the atmospheric terms, and then to estimate band emissivity ratios. The radiance difference can be expressed as ARi - - -
Ril
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= r,e,[B,(Tx) - B,(T2)] + O,,
(5)
w h e r e O~ = (1 - e 0 [ n s k y , - Hsky~2] + [Hpath~ Hpathi2] is an atmospheric term that can be estimated, and ri is assumed temporally invariant. A linear regression can be c o m p u t e d between the observed value ARi and an estimate of T1- Tz. The intercept of this regression determines the value of Oi. Because this plot is constructed, the fit of the linear regression to the plotted data provides a test of whether this procedure provides a satisfactory estimate of the O,s. The emissivity ratios c o m p u t e d from Eq. (5) give a numerically stable form: e,/e,+~ = (AR~I AR~, ~)(r,, 1/ r,) x [IB, +~(T~)- B,, ~(T~)] //B,(T~) - B,(T2)}], (6) where AR~ - A R i - Oi. The radiances R~ are comp u t e d directly from the scanner data after applying the appropriate calibration constants (Palluconi and Meeks, 1985). The offsets O~, as described previously, are computed from the intercepts of regression lines. The atmospheric
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transmission ratio can be measured directly or estimated assuming gray body emissivities for the areas of densest vegetation cover in the scene. The third term in the expression, which involves ratios of blackbody radiance differences, is comp u t e d using estimates of the temperatures. Because this expression is independent of temperature to first order, the approximation is quite satisfactory. The ratio algorithm [Eq. (6)] has been tested using the radiance simulation model in a similar fashion to the previous tests for emissivity. The results are comparable, but there is a signal:noise increase of about 20 for the ratio. This increase is in agreement with a simple theoretical model, indicating that the signal:noise change should be approximately proportional to ~/A;t (for TIMS this is approximately 10 / 0.5 or 20). We conclude from these simulation studies that subtle compositional differences, which are reflected in spectral shape changes, can be detected using emissivity ratios with a signal:noise that is comparable to that of the TIMS instrument. REFERENCES Gillespie, A. R. (1980), Digital techniques of image enhancement, in Remote Sensing in Geology (B. R. Siegal and A. R. Gillespie, Eds.), John Wiley, New York, pp. 139226. Gillespie, A. R., Kahle, A. B., and Walker, R. E. (1986), Color enhancement of highly correlated images. I. Decorrelation and HSI contrast stretches, Remote Sens. Environ. 20:209-235. Hummer-Miller, S. (1990), Noise removal and registration of TIMS remote sensing aircraft data, Photogramm. Eng. Remote Sens. LVI(1):49-53. Palluconi, F. D., and Meeks, G. R. (1985), Thermal Infrared Multispectral Scanner (TIMS): an investigator's guide to TIMS data, Jet Propulsion Lab. Publ. 85-32, Pasadena, CA. Schowengerdt, R. A. (1983), Techniquesfor Image Processing and Classificationin Remote Sensing, Academic,New York, 249 pp. Watson, K., Kruse, F., and Hummer-Miller, S. (1990), Thermal infrared exploration in the Carlin Trend, Geophysics 55(1):70-79.