274 
True) = Tote SXP(-Atuet) + Spwuell) (2) 
where ere is à random variable with zero mean and standard deviation Of £F true ANd Eptrue IS à random variable with 
zero mean and standard deviation of Ep (See Table 1). 
The gain of the detector also was assumed to change with time, but it was not one of the state variables in this Kalman 
Filter. The following equation, based on analysis of data from the detectors in band 1 of the Thematic Mapper sensor on 
Landsat 5, was used to describe how the gain of the detectors in band 1 changed as a function of time: 
Glt,+1) 7 G(t)-0.0000444227 -* eju, with G(t) = 1 (3) 
where @gain is a random variable with zero mean and standard deviation of Ep (see Table 1). The data generated using 
Equations (1) through (3) was used to evaluate the output of the Kalman Filter, as well as to provide input to the Kalman 
Filter in the form of simulated measurements. 
The truth model provided simulated FASC, PASC, and GLC measurements (Dg, Dp, and Dg output detector counts) to 
the filter, at the frequency specified in Section 3 of this paper, using the following equations: 
Dalt) = G(t)Ra(t) (4) 
Delt) = Glt)owue(td Ree (te) (5) 
Delt) = Glt)True(td Rept) (6) 
where o and : were as specified in Equations (1) and (2), G was as specified in equation (3) and Ra, Rer, and Rgp , the 
true GLC radiance, true full aperture solar radiance, and true partial aperture solar radiance, were computed using the 
following equations: 
Relt) = Ra + ea(ty : (7) 
Rge(t) ^ Rae + slt) (8) 
Rep(ti) = Rgp + Esplti) (9) 
In equations (7) through (9), ea, esr, and esp are random variables with zero means and standard deviations of eq tue, 
ESF,true; 8d £sp.rue, respectively (see Table 1), and “barred” variables are mean radiance values. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
PARAMETER TRUE MODELED 
o (INITIAL) 0.9 0.86 
1 (INITIAL) 0.06 0.054 
€ sr (solar variability for FASC) 0.18% 0.18% 
£ sp (solar variability for PASC) 0.1896 0.18% 
€ ç (randomness in FASC) 1.7% 1.7% 
€ p (randomness in PASC) 0.02% 0.02% 
€ a (randomness in true GLC 0.096 0.096 
radiance) 
E r (detector gain drift at FASC) 0.296 0.296 
E p (detector gain drift at PASC) 0.296 0.296 
1| a (randomness in modeled GLC 2.596 2.596 
input radiance) 
P a (bias in GLC radiance) 5.096 0.0% (estimated from data) 
a 0.000028* 0.0001** 
b Asymptotic*** 0.0000577**** 
  
  
  
  
Table 1. Initial Values and Noise Statistics Assumed for Kalman Filter and Truth Model 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995