Olsson, Hakan 
realistic surface reflectances. This situation may improve with future sensors, but there are still advantages to using 
purely image-based statistical methods for relative image calibration in this context. 
In response to the difficulties with absolute calibration and atmospheric correction, several image-based statistical 
techniques have been used. These usually make use of the observation that for a narrow field-of-view sensor such as 
Landsat TM, recorded digital counts are a near-linear function of surface reflectance within each spectral band. Even 
the physics-based calibration procedures arrive at this result after simplifying assumptions are made. Thus rather than 
estimating linear calibration factors to convert to surface reflectance, you can estimate linear calibration factors to 
compensate for the relative scale difference from one scene to another. These procedures normally make use of a set of 
stable reflectance targets or so-called pseudoinvariant features (Schott and Salvaggio, 1988, Heo and FitzHugh, 2000), 
or bright and dark control sets derived from image scattergrams (Hall et. al, 1991). However, for forest monitoring, 
slightly better results can be achieved by using a selection of forest pixels for the ‘stable’ reference (Olsson, 1993). 
This has the effect of also neutralizing other effects such as vegetation phenology and sun-angle effects, at least to the 
best linear approximation. 
Here we use a method that was described in Joyce and Olsson (1999). It consists of deriving a band-specific linear 
correction factors between each pair of adjacent images, and propagating this correction through the sequence to match 
a specified reference. We used the 1994 image as a reference. This method achieves good results because the 
correction factors are derived from image pairs that are close together in time, and it doesn't rely on having complete 
overlap of all images for all dates. It does not however compensate for the expected drift in the mean value over time. 
3.2 Temporal data analysis 
The methods for detecting unexpected changes between two dates of imagery are well developed and tested (see Coppin 
and Bauer, 1996 for a review) but there have been few attempts to use longer time series of images to monitor gradual 
trends such as those associated with forest growth or decline. Since these are clearly time-dependent data, the first 
thought may be to apply some of the well-developed theory of analysis of time series to model individual plot spectral 
trajectories. The problem is that these time series are too short to derive estimates of autocorrelation, the time points 
may not be equally spaced, and there may be frequent missing values. One could also fit individual regression models 
to each short sequence without regard to the serial correlation and use the coefficients of the regression models to make 
inferences. Lawrence and Ripple (1999) use this technique for monitoring vegetation recovery based on temporally 
modeled crown closure estimates. 
There are techniques applied in the social sciences in the field of longitudinal data analysis (see Diggle, 1994) that may 
be useful in this application. A longitudinal data set consists of time sequences of measurements taken from several 
experimental units (called Subjects from the social science heritage). The subjects can be regarded as a sample from 
some underlying population, and often have covariate information attached to them. The focus of longitudinal studies is 
to compare the differences in temporal behavior between subjects or groups. In this context, our subjects are sample 
plots and the covariate information may be site index or species composition. The observed response variable over time 
is the spectral values from a normalized image time sequence. 
Longitudinal analysis is in contrast to cross-sectional analysis where temporal behavior is inferred by comparing 
experimental units with different ages at a common time. We start the data exploration by looking at the relationship 
between spectral values and age. This gives an idea of the expected time profile of an individual plot, however there are 
potentially large differences between conditions on plots that aren't controlled for. The advantage of longitudinal 
analysis is that the between-plot differences are separated from the temporal behavior of individual plots. 
Statistical analysis was performed with the Oswald extension to Splus (Smith et al, 1996), which is available for free 
from the University of Lancaster. Oswald extends SPlus data types to include a longitudinal data frame, or collection 
of time series indexed by subject. Additional covariates can be added to the data frame for use in regression models. 
These covariates can be subject-specific (different values for each subject, but constant over time), or time-specific 
(different value for each time, but constant over subjects). 
4 RESULTS 
4.1 Cross-sectional data exploration 
We start by a cross-sectional data exploration of the 1994 image. Since the field data are collected on different dates, 
the age is updated to the year 1994. Age is actually basal-area weighted age on the plot, except in the very young stands 
where the stand age is used.  Scatterplots of spectral values in band 4 and 5 v.s. age are shown in figures 3a and 3b. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000. 1085