376 
time t+1 in equation (34), given its estimated 
status Jft at time t, is analogous to the 
calibrated estimate in equation (7), given an 
imperfect (biased) remotely sensed estimate. 
If the deterministic model is perfect, variance 
vari^t + i) of estimated status Jft + i of the one 
sample unit at time t+1 is 
var(Jt + i) = var( Xt) Pt 2 + var(Jft) A 2 , (35) 
as portrayed in Fig. 2. More realistically, the 
deterministic model is imperfect, and there is 
additional error (Ut) in predicting change 
between time t and t+1. Assuming additive, 
independent prediction errors, variance of the 
updated estimate for the one sample unit is 
var(2it*i) = var(lft )ft 2 +var(Xt) A 2 +var( Ut), (36) 
as portrayed in Fig. 2. One fundamental problem 
will be estimating the variance of the prediction 
errors va.r(Ut) between times t+1 and t. This is 
discussed in Section 7.3. 
7.2 Stratum estimates for each time period 
The stratum level prediction model (i.e., 
transition probabilities Pi and A) in equations 
(34) and (36) could update the estimated status 
of each sample unit in the 1/4 subsample observed 
at time t. These estimates might be directly 
combined with those from the other 1/4 subsample 
observed at time t+1, using the composite method 
presented in Section 5. The resulting stratum 
estimate at time t+1 would include measurements 
from 1/2 of the sample units. 
At time t+2, the estimated status of each sample 
unit in the 1/4 subsample observed at time t+1, 
and the 1/4 subsaaple observed at time t and 
updated to time t+1 using equations (34) and 
(36), could be updated to time t+2 using 
Xt *2 = Pt Xtti + Po (1-Iui), (37) 
var(lt« 2 ) = ft 2 var(At 4 l) + A 2 var(lt 4 l )+var( Ut 4 l). 
(38) 
These updated estimates from the 1/4 subsamples 
observed at times t+1 and t might be directly 
combined with those from the 1/4 subsample 
observed at time t+2, using the composite method 
in Section 5. The resulting stratum estimate at 
time t+2 would include measurements from 3/4 of 
the sample units. 
The same method might be applied at time t+3 to 
estimate stratum status using all sample units. 
Most weight in the composite estimator would be 
placed on the 1/4 subsample observed at time t+3 
because a prediction model is not needed to 
update estimated status of sample units within 
this subsample, and there would be no prediction 
errors; least weight would be placed on the 
subsample observed at time t because their status 
has not been directly observed for 4 time 
periods, and variance from prediction errors in 
updating estimated the status of the sample units 
would be greatest for this 1/4 subsample. 
7.3 Variance of prediction errors 
Variance of prediction errors from the 
deterministic model , i.e., var(Dt) = var(i/t4i) = 
var(U), are needed in (36) and (38) to update 
status estimates for sample units, which are 
combined into an estimate for the stratum. The 
variance of sampling errors from transition 
probabilities estimated using permanent ground 
plots (from other agencies or more detailed field 
sampling within the same monitoring system) might 
serve as initial estimates of prediction error 
variance. Initial estimates of prediction error 
variance for a process level landscape model 
might be made with data used to fit the model. 
These initial estimates are likely biased (i.e., 
too small) because the deterministic model is 
extrapolated over time or space. Stratum 
estimates from Section 7.2 can be compared to 
independent stratum estimates from other 
monitoring systems, and the adaptive methods 
discussed in Section 6.1 used to refine estimates 
of prediction error. 
Direct estimates of prediction error variance 
from the deterministic model would be available 
through remote sensing of permanent sample units. 
For example, new imagery is acquired at time t+4 
for the same 1/4 sample observed at time t. 
Misclassification bias in the estimated status of 
each sample unit at time t+4 is corrected using 
the calibration model in Section 3. A second 
estimate of the status of each sample unit in the 
1/4 subsample at time t+4 is available from the 
deterministic prediction model, using the 
observed status at time t as initial conditions 
(Section 7.2). A sample estimate for variance of 
prediction errors between times t and t+4 can be 
made using the known differences between these 
two estimates at time t+4 for each sample unit. 
The remotely sensed estimate of these sample 
units at time t+4 would then be used as new 
initial conditions in the deterministic model to 
predict status at time t+5 and later. 
This requires matrix representation of the 
statistical model, as in equation (25). The 
matrix solution for estimating var(U) would be 
complicated by covariances among prediction 
errors, use of the same calibration model at 
times t and t+4, or spatial autocorrelations. 
Approximations might be needed, but verification 
procedures introduced in Section 6.1 could 
protect against unreliable approximations. 
8. KALMAN FILTER APPLIED TO CELLS 
Each 40 km 2 sample unit may be considered a 
sample of the surrounding 640 km 2 cell, with a 
sample size of one. Estimates for aggregations 
of cells might utilize composite estimation 
(Section 5), treating the estimate for a 40 km 2 
sample unit as an estimate of the entire 640 km 2 
cell. This can reduce proliferation of 
stratification criteria from the calibration 
models and deterministic prediction models, and 
use of ancillary estimates from independent 
sources. 
8.1 Combining independent ancillary estimates 
Ancillary statistical estimates from independent 
sources can improve efficiency and temporal 
detail using composite estimation. For example, 
the USDA Forest Service and the USDA Soil 
Conservation Service both produce areal estimates 
of the extent of forestlands for geographic areas 
that might include one-hundred or more 640 km 2