The Kalman filter is usually applied to a time 
series of measurements (Fig. 3). With each new 
measurement, a composite estimate is made, whicl 
serves as new initial conditions for the next 
prediction from the deterministic model (e.g., 
Fig. 3, year 4). 
c 
<D 
O 
<5 
CL 
No 
monitoring data 
Low precision 
monitoring data 
Kalman composite 
Estimate from 
prediction model 
disagrees with 
measurement 
Smaller 
prediction error 
Composite 
estimate 
Other 
estimate 
Model 
estimate 
Increasing estimate 
of prediction error 
improves 
agreement 
Larger 
prediction error 
Fig. 4 Expected probability densities for two 
estimates that disagree. Adaptive filters assume 
the estimated variance of model prediction error 
is inaccurate, and change this estimate until the 
disagreement is within acceptable bounds. 
Fig. 3 Kalman estimates and confidence intervals 
for percent forest. In this example from 
Czaplewski, et al. (1988), intensive forest 
inventories were conducted in years 0 and 10; 
lower precision monitoring data were gathered in 
years 4 and 7. 
The Kalman filter is a multivariate estimator 
(Maybeck 1979). It can simultaneously estimate 
multiple state variables, such as proportions of 
different vegetation cover types. Measured rates 
of change can be statistically combined with 
rates of change predicted from the deterministic 
model. The Kalman filter can model correlated 
errors among the state variables and rate 
coefficients, correlated prediction errors from 
the deterministic model, and random errors in 
measurement data. 
6.1 Verification of the Kalman filter 
Two independent estimates disagree or "diverge" 
in that neither estimate is likely given the 
other (Fig. 4). Contradictory estimates can be 
combined, but the resulting composite estimate 
can be biased. Discrepancies are can be caused 
by biased estimates of the error distribution 
(either location or spread) of the measurement at 
time t, or the estimate at time t-1 that is 
updated to time t using the deterministic 
prediction model. 
It is possible that bias exists in the current 
measurement. For example, calibration equations 
are needed to correct for misclassification bias, 
as discussed in Section 3. Also, bias might 
exist in the estimated variance of errors from 
the prediction model; direct estimates of 
prediction variance require known differences 
between model predictions and the true status of 
the system. As an alternative, adaptive filters 
modify initial variance estimates until 
disagreements are within acceptable bounds (Fig. 
4), often using a time series of residuals 
(Sorenson 1985). As accuracy of model 
predictions increases, the weight placed on model 
predictions will increase, and as will accuracy 
of the Kalman filter. 
7. KALMAN FILTER APPLIED TO SAMPLE UNITS 
Consider the following hypothetical example, in 
which a 1/4 subsample of the sample units in the 
stratum are observed using remote sensing with 
imagery acquired at time t in an interpenetrating 
design. (Similar examples could be based on 
other intensities, such as 1/7, 1/9, 1/12, etc.) 
The biased estimate of the status of each sample 
unit in the 1/4 subsample is corrected using a 
stratum level calibration estimator, as in 
Section 3. An estimate of stratum status at time 
t is made with the 1/4 subsample using composite 
estimation (Section 5). A different 1/4 
subsample of sample units is observed using 
remote sensing and imagery acquired at time fc+1. 
An estimate of stratum status at time fc+1 might 
be made using only this second subsample, as 
described in Sections 3 and 5. 
The estimate for time fc+1 might be improved using 
the sample units in the first 1/4 subsample, 
which were observed at time fc. However, changes 
between times fc+1 and fc have probably occurred in 
the status of each sample unit in the first 1/4 
subsample. If a model were available to predict 
these changes, then estimates from the 1/4 
subsample observed at time fc might be combined 
with the 1/4 observed at time fc+1 into an 
estimate of the stratum status at time fc+1, using 
the composite estimator presented in Section 5. 
7.1 Updating estimates for one sanple unit 
Predicted true status Jft*i (e.g., proportion 
forest) of one sample unit at time t+1 is 
Xt*i = ft Xt + fb (1-Xt). (34) 
Xt is the estimated status of the sample unit at 
time t, ft is the estimated conditional 
transition probability that a point is truly 
forest at time fc+1, given it is was forest at 
tine fc, and ft is the estimated conditional 
transition probability that a point at time fc+1 
is truly forest, given it was other cover at time 
fc. Transition probabilities ft and ft are 
predicted from the deterministic model. The 
predicted status Xt*i of the one sample unit at 
375