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Plot Collocation Error: Impacts on
Area Estimation
Willem W. S. van Hees
'
Abstract--Results of a study conducted to examine area estimation error
caused by improper collocation of aerial photo plots and associated
ground plots are presented. Analyses considered the complexity of the
land cover pattern on the plot, cardinal direction of collocation error, and
percentage of area by land cover class on control plots.
During the 198O's, the Forest Inventory and Analysis (FIA)-Anchorage
unit of the Pacific Northwest Research Station conducted renewable
resource inventories employing a three-phase-with-subsampling
inventory design. There were three remotely sensed samples and one
ground sample. Regression estimators were developed for several
resource quantities, particularly area by land cover class. Results from
two inventories showed low correlations between most covariates on the
different sampling layers. Statistical consultation indicated inaccurate
collocation of sample plots could contribute to poor correlations.
This study focused on the effects of improper collocation between low
altitude aerial photo plots and ground plots on estimation of productive
forestland area. Results indicate there may be a north-south directional
bias and that land cover complexity as estimated by fractal dimension is
an important interaction component. Area estimation errors ranged from
zero for homogeneous plots to 10.9 percent for plots with more complex
land cover patterns.
INTRODUCTION
During the 1980's the Anchorage Forest Inventory and Analysis (FIA) unit of
the Pacific Northwest Research Station, conducted experimental multiresource
inventories employing a three-phase sampling with subsampling design
(Schreuder, Gregoire, Wood 1993). Landsat multispectral imagery (LS), highaltitude color infrared photography (HAP), low-altitude color infrared
Research forester, Pacific Northwest Research Station, Anchorage, AK.
photography (LAP), and ground (G) plots were the sampling layers. A grid of 8
ha plots on the LS layer was sampled with successively more extensive subgrids
on HAP, LAP, and G. The subgrid of LAP plots was subsampled on the ground.
At the LS layer the grid spacing was 5 km, at the HAP layer 10 km, 20 km at the
LAP layer, and 40 krn on the ground, Figure 1.
At each sample location land
cover class was evaluated.
Land cover polygons were
mapped for area estimation at
remotely sensed locations and
were point sampled at ground
locations.
Population totals
were estimated using several
estimators, including regression
estimation. Regression results
were disappointing in that
correlations between covariates
and independent variables were
not satisfactory. A number of
factors, including unrecognized
sources of variation and
changing
objectives
were
deemed responsible.
A
possible source of uncontrolled
variation was plot collocation.
I
Figure 1--Three-phasesampling with subsampling
For regression estimators in
grid design used for multi-resource inventories in
particular, plots at various
Alaska during the 1980's.
sample stages
must
be
accurately collocated.
This
study was designed to investigate the hypothesis that apparently minor plot
collocation errors can significantly contribute to area estimation error.
I
Two factors that could contribute to relative error magnitude were considered; the
amount of a given land cover class on the plot and the complexity of the land
cover pattern on the plot. If a plot is homogenous with respect to land cover class,
then small errors in collocation would, at best, result in small relative errors in
area estimation. On the other hand, if the plot has small amounts of a given land
cover class, then small collocation errors could result in larger relative area
estimation errors. To represent land cover class complexity, the fractal dimension
of the mapped land cover pattern within the plot was estimated. The hypothesis
then, was that area estimation error for a given land cover class (expressed as the
difference in percent of plot area in that land cover class on a control plot versus a
mislocated plot) would be a function of the total plot area in that land cover class
(expressed as a percent of total plot area) and of the overall fractal dimension of
the land cover pattern on the plot.
METHODS
Study Area
Field data for this study did not exist; this study is a simulation in so far as
production of collocation errors is concerned. Aerial photos from inventories of
the central interior, east-central interior, and south central coastal regions of
Alaska (Figure 2.) were used as the data source.
Data Collection
fi
Alaska
Color infra-red aerial photos
of 3 1 different locations were
subjectively
selected
to
represent
a
variety
of
topographies,
latitudes,
longitudes, and land cover
classes. Aerial photo scales
ranged from 1:3,000 to
1:6,000.
Figure 2--Study plots were selected from within
the shaded area above.
An aerial photo interpreter
classified and delineated, on
transparent overlays, up to 4 land cover classes on each photo. The entire photo
was classified. Classes were chosen to simplify the classification process and
thereby reduce interpretation error. The classes were productive forestland
(forestland capable of producing 1.4 m3.ha-leyr-'), other forestland, nonforest, and
water .
Each of the resulting 31 land cover class map overlays was digitized. On the
digitized images 8 ha control plots were drawn. These plots served as the source
of the true, or ground, area. For each control plot, area by land cover class was
measured and converted to a relative value (P,).
Improper collocation between these control (or ground) plots and their
associated aerial photo plots was modeled by overlaying new plots that were offcenter. For each control plot, improperly collocated simulated aerial photo plots
were created by overlaying, off-center, new plots moved in one of eight cardinal
directions. Directions of movement were randomly chosen without replacement.
The amount of collocation error was approximately 0.65 cm on 1:6,000 scale
photos (approximately 40 m on the ground). The magnitude of this error derives
from field experience gained during the course of the inventories. Five such
mislocated plots were established for each control plot. For each mislocated plot,
area by land cover class was also measured and converted to a relative value (P,).
Sets of control and moved plots were created until there were at least 5
observations in each cell of frequency tables for each land cover class. The
frequency tables listed number of plots by complexity class (simple vs complex)
and percentage of the control plot in the land cover class. The percentages of the
control plot in a given land cover class were grouped into quarters. In order to
place each plot in a complexity class, estimates of the fractal dimension of the
land cover pattern on control plots were made using the dividers method
(Sugihara and May 1990). Complexity class was established according to a
vegetation complexity model for Alaska (van Hees 1994). Ultimately, 195 sets of
control plots and associated moved plots were used for this study.
Data Analysis
For each set of control and moved plots, the differences in percent of area (or
area estimation error), by land cover class, between the moved and control plots
were found. The mean of the absolute values of the five differences in percent of
area (Pe) was then calculated as:
where:
PCand P, as defined above.
For preliminary analysis, the GLM (General Linear Models) procedure (SAS
1990a) was used to fit a linear regression model, by land cover class, to the data.
The model used was:
where:
Pe = mean absolute area difference,
Po = intercept parameter,
P1,2,3 = regression coefficients,
1 < Fd < 2
Fd = fractal dimension.
To investigate the possibility that directional bias existed, the same general
linear model as in (2) was used except that individual absolute difference for the
particular direction of mislocation was the dependent variable rather than mean
absolute difference over the five mislocated plots.
Subsequent to linear analyses, cluster analyses were conducted using the
FASTCLUS procedure (SAS 199Ob). Using FASTCLUS, observations are
divided into clusters on the basis of Euclidean distances computed from one or
more quantitative variables. For this study the variables used for clustering were
Pe, Pc, Fd, and (PC * Fd).
RESULTS
Initial screening by linear regression showed significant coefficients for the
productive forestland class only. T-values and analysis of variance statistics are
presented in Tables 1 and 2. The predictive value of the regression relation
however, as measured by 3, was negligible. The value for this relationship was
0.129.
Table 1--T-tests of significance of individual regression coefficients for multiple
linear regression analysis of P, against PC,fractal dimension (Fd),and PC*Fd.
Variable
Coefficient
Intercept
0.2244931
-0.3542443
-0.1525531
0.3058221
PC
Fd
pc*Fd
Standard error
0.0579952
0.1019977
0.048 1601
0.0886174
P (two-tail,
T
3.87
-3.47
-3.17
3.45
0.0002
0.0008
0.0021
0.0009
Table 2--Analysis of variance for multiple linear regression analysis of P,
against PC,fractal dimension (Fd), and PC*Fd.
Source
Model
Error
Total
DF
3
84
87
Sum of squares
Mean square
F value
0.00942545
0.06352763
0.07295308
0.00314182
0.00075628
4.15
Pr > F
0.0085
Adding a squared interaction term so that the model became
Pe =
PO+ PI ( Pc ) + P 2 ( F d ) + P 3 ( PC * F d ) + P 4 ( Pc * F d )27
(3)
improved predictive power. The rL value for the expanded model was 0.375.
Again, the productive forestland class was the only class for which all regression
coefficients were significant.
The mean difference (P,) between the percentages on the control plots versus the
mislocated plots ranged from zero to 10.9 percent with a mean of 4.3 percent and
standard deviation of 2.9 percent. In the case where there was zero difference
between the control and the mislocated plots all plots were homogenous. Land
class pattern fractal dimension (Fd) ranged from 1.000 for homogenous plots to
1.530 for the most complex patterns. Mean Fd was 1.165 with a standard
deviation of 0.109.
Model (3) was used to examine the data for possible directional bias. That is,
could the direction in which mislocation of the aerial photo plot occurred affect
the magnitude of relative area estimation error. Table 3 presents the results of this
analysis. Noticeable is the strength of the estimated regressions for those
observations where the direction of mislocation was along the northhortheast by
south/southwest axis (highlighted figures in Table 3) when compared with the
estimates for other directions.
Table 3-F-values and T -scores for estimated coefficients of model (3) by direction of
movement of improperly collocated plots for productive forestland.
T for Ho = 0
Direction
North
Northeast
East
Southeast
South
Southwest
West
Northwest
P C
~d
-2.99
-2.59
-1.92
-1.81
-3.31
-1.74
-1.77
-1.12
-2.70
-2.61
-1.32
-1.21
-2.53
-1.80
-1.50
-0.77
P C
* Fd
3.70
3.30
2.53
2.28
4.04
2.83
2.01
1.08
(PC
* F d )2
-4.37
-1.83
-3.29
-1.68
-3.96
-2.87
-2.08
-1.16
r2
.3011
.2767
.I706
.I388
.2898
,3526
.0824
.I359
Clustering of the data was undertaken to examine the possibility that the
predictive power of the model (using equation (1)) could be improved.
FASTCLUS was run setting the number of clusters at 3 and 4. Overall r2 for the
4-cluster grouping was higher than for the 3-cluster grouping (0.886 vs 0.829).
Clusters were separated, by FASTCLUS, into the following ranges of PC with
almost no overlap. The groups of PCwere 0 - 15% (n=18), 15% - 45% (n=20),
45% - 75% (n=31), and 75% - 100% (n=19). Model (2) was then used to examine
the within cluster regression relationships. In the first and last clusters the
regression relationships were quite strong (r2 = 0.729 and 0.841 respectively).
The relationship for the second cluster was much weaker (r2 = 0.385) and for the
third cluster the regression relationship was essentially nonexistent (r2 = 0.026).
DISCUSSION
With regards to the productive forestland component of the study area, the
hypothesis that area estimation error would be a function of the percentage of the
control plot (PC)in a land cover class and the fractal dimension of the land cover
pattern is supported by the results of this study.
Although this study does not provide substantive clues as to why the model was
more informative for productive forestland than for other land cover classes a
possible explanation lies in the physiography of the study area and the discovery
of possible directional bias. Patterns of productive forestland tend to be linear in
the study area. In interior Alaska productive forestland is found along rivers on
south facing slopes whereas in coastal regions it is "marginal" - that is along the
margins next to water and up to about 600 m above sea level. Rivers and water
bodies in the study area have a predominantly east/west orientation. Thus, plot
mislocations to the north or south would likely have more impact on error
magnitude than mislocations to the east or west.
Clustering of the observations did improve predictive ability for certain ranges
of the data; particularly the upper and lower ends of the control plot percentage
scale. It would be possible then, to more accurately assess impacts of rnislocation
knowing the percentage of the plot in a land cover class along with an estimate of
fractal dimension of the mapped land cover classification.
Although the results of this study do not provide strong modeling capabilities
they do provide useable guidelines. It is apparent that small mislocations between
aerial photo and ground plots can produce significant area estimation errors (up to
10 percent) and that inventory designers and data analysts must be aware of
physiography in order to consider possible introduction of directional bias. Also,
there may be some predictive capability for extremes of land cover percentages.
LITERATURE CITED
SAS. 1WOa. SAWSTAT User's guide, Version 6, Fourth ed., vol. 2. SAS
Institute, Inc., Cary, NC. USA.
SAS. 1990b. SAWSTAT User's guide, Version 6, Fourth ed., vol. 1. SAS
Institute, Inc., Cary, NC. USA.
Schreuder, Hans T., T. G. Gregoire, and G. B. Wood. 1992. Sampling methods
for multi-resource forest inventory. John Wiley and Sons, Inc., 605 Third Ave.,
New York, New York. 10158-0012. 446 p.
Sugihara, George & May, Robert M. 1990. Applications of fractals in ecology.
Trends in Ecology and Evolution, Elsevier Science Publishers Ltd., (UK), 5(3):
79-86.
van Hees, Willem W. S. 1994. A fractal model of vegetation complexity in
Alaska. Landscape Ecology. 9(4):27 1-278.