Quick Review of Remote Sensing Basic Theory Paolo Antonelli CIMSS University of Wisconsin-Madison Benevento, June 2007

Outline Planck Function Infrared: Thermal Sensitivity Bit Depth

Spectral Distribution of Energy Radiated from Blackbodies at Various Temperatures

Radiation is governed by Planck’s Law In wavelenght: B(,T) = c 1 /{ 5 [e c2 / T -1] } (mW/m 2 /ster/cm) where = wavelength (cm) T = temperature of emitting surface (deg K) c 1 = x 10-8 (W/m 2 /ster/cm -4 ) c 2 = (cm deg K) In wavenumber: B(,T) = c 1 3 / [e c2 /T -1] (mW/m 2 /ster/cm -1 ) where = # wavelengths in one centimeter (cm-1) T = temperature of emitting surface (deg K) c 1 = x 10-5 (mW/m 2 /ster/cm -4 ) c 2 = (cm deg K)

B( max,T)~T 5 B( max,T)~T 3 B(,T) B(,T) versus B(,T) wavelength [µm] max ≠(1/ max )

wavelength : distance between peaks (µm) Slide 4 wavenumber : number of waves per unit distance (cm) =1/ d =-1/ 2 d

Using wavenumbers Wien's Law dB( max,T) / dT = 0 where (max) = 1.95T indicates peak of Planck function curve shifts to shorter wavelengths (greater wavenumbers) with temperature increase. Note B( max,T) ~ T**3.  Stefan-Boltzmann Law E =   B(,T) d =  T 4, where  = 5.67 x 10-8 W/m 2 /deg 4. o states that irradiance of a black body (area under Planck curve) is proportional to T 4. Brightness Temperature c 1 3 T = c 2 /[ln( ______ + 1)] is determined by inverting Planck function B Brightness temperature is uniquely related to radiance for a given wavelength by the Planck function.

Using wavenumbersUsing wavelengths c 2 /T c 2 / T B(,T) = c 1 3 / [e -1]B(,T) = c 1 /{ 5 [e -1] } (mW/m 2 /ster/cm -1 )(mW/m 2 /ster/cm) (max in cm-1) = 1.95T (max in cm)T = B( max,T) ~ T**3. B( max,T) ~ T**5.  E =   B(,T) d =  T 4, o c 1 3 c 1 T = c 2 /[ln( ______ + 1)] T = c 2 /[ ln( ______ + 1)] B 5 B

Temperature sensitivity dB/B =  dT/T The Temperature Sensitivity  is the percentage change in radiance corresponding to a percentage change in temperature Substituting the Planck Expression, the equation can be solved in  :  =c 2 /T

∆B 11 ∆B 4 ∆B 11 > ∆B 4 T=300 K T ref =220 K

∆B 11 /B 11 =  11 ∆T/T ∆B 4 /B 4 =  4 ∆T/T ∆B 4 /B 4 >∆B 11 /B 11   4 >  11 (values in plot are referred to wavelength)

∆B/B=  ∆T/T Integrating the Temperature Sensitivity Equation Between T ref and T (B ref and B): B=B ref (T/T ref )  Where  =c 2 /T ref (in wavenumber space) (Approximation of) B as function of  and T

B=B ref (T/T ref )   B=(B ref / T ref  ) T   B  T  The temperature sensitivity indicates the power to which the Planck radiance depends on temperature, since B proportional to T  satisfies the equation. For infrared wavelengths,  = c 2 /T = c 2 / T. __________________________________________________________________ Wavenumber Typical Scene Temperature Temperature Sensitivity

Non-Homogeneous FOV N 1-N T cold =220 K T hot =300 K B=NB(T cold )+(1-N)B(T hot ) BT =N BT hot +(1-N)BT cold

For NON-UNIFORM FOVs: B obs =NB cold +(1-N)B hot B obs =N B ref (T cold /T ref )  + (1-N) B ref (T hot /T ref )  B obs = B ref (1/T ref )  (N T cold  + (1-N)T hot  ) For N=.5 B obs /B ref =.5 (1/T ref )  ( T cold  + T hot  ) B obs /B ref =.5 (1/T ref T cold )  (1+ (T hot / T cold )  ) The greater  the more predominant the hot term At 4 µm (  =12) the hot term more dominating than at 11 µm (  =4) N 1-N T cold T hot

Consequences At 4 µm (  =12) clouds look smaller than at 11 µm (  =4) In presence of fires the difference BT 4 -BT 11 is larger than the solar contribution The different response in these 2 windows allow for cloud detection and for fire detection

Low Gain Channels Band 14 low 0.68 µm Vegetated areas Are visible Saturation over Barren Soil Visible details over water

High Gain Channels Band 14 hi 0.68 µm Saturation over Vegetated areas little barely visible Visible details over water Saturation over Barren Soil

Range for Band 14 high 0.68 µm Range for Band 14 low 0.68 µm

Bit Depth and Value Range With 12 bits 2 12 integer numbers can be represented –Given ∆R, the range of radiances we want to observe, the smallest observable variation is ∆R/ 2 12 –Given dR smallest observable variation, the range of observable radiances is dR* 2 12 For this reason Band 14low (larger range) is used for cloud detection and Band 14hi (smaller range) is used for ocean products ∆R dR

Conclusions Planck Function: at any wavenumber/wavelength relates the temperature of the observed target to its radiance (for Blackbodies); Thermal Sensitivity: different emissive channels respond differently to target temperature variations. Thermal Sensitivity helps in explaining why, and allows for cloud and fire detection; Bit Depth: key concepts are spectral range and number of bits.