In this study, the comparisons of the proposed plot design I, II and III with their nonadaptive counterparts based solely on their statistical performances without considering costs.
Adaptive plot designs demand more time in the field and thus cause higher cost than fixed-area plot designs for a given sample size. But in reality, the main concern in forest inventories especially large area forest inventories is quite often which sampling strategy can produce an estimate of the best precision for a given cost. A sampling strategy which is superior for a given number of elements in the sample may yield a less precise estimate for a given cost. In other words, it may not be cost efficient. One typical relevant example is SRS and conventional random cluster sampling. For a given number of elements in the sample, SRS is usually known to have a better statistical performance than conventional cluster sampling, but for a given cost a more precise estimate may be obtained with conventional cluster sampling under certain conditions (Cochran, 1977; Matérn, 1986; Thompson and Seber, 1996) since travelling to a new random sample point in an inventory region is generally much more time-consuming than the observations on a cluster of subplots installed at an existing sample point in cluster sampling. Thus, in practice overall efficiency, which takes costs into account, is mostly the decisive factor for the choice of sampling strategy. For this reason, the overall efficiencies of the three proposed adaptive plot designs need still to be compared for their practical application, although they are akin to conventional cluster sampling in nature and the
57 results obtained in this study are positive for them. A comparison of overall efficiency requires cost functions based on the assumptions about the additional field effort.
The cost functions here are developed from the modification of a cost function for SRS which is simplified from Scheuber and Köhl (2003). All costs involved in the following cost functions are expressed in terms of time, not money. The notations in all cost functions are used consistently. The cost function for SRSis:
where:
total cost in terms of time,
average time requirement of the round travel between the camp and the inventory region, average time requirement to travel to next sample point,
average time requirement for field observation within a sample plot, and sample size.
3.2.1 Plot design I
The cost function for plot design I depends not only on its design factors but also on the estimator for it. When the estimator presented in this study is used, the collection of the coordinates of all sampled trees as well as their relevant trees and its cost must be accordingly added into the cost function as a new cost item. The cost function for plot design I modified from CT is then:
[ √ ] [ (√ ) √ ] where:
total cost in terms of time for plot design I,
average time requirement of the round travel between the camp and the inventory region, average requirement to travel to next sample point,
average time requirement for field observation of the target variable per initial sample plot area,
average time requirement for collection of coordinates of relevant trees of the sampled trees per initial sample plot area,
58 sample size or number of initial sample plots,
expected value of the number of initital plots with the presence of at least one tree, expected value of the number of initial plots with the presence of at least i trees (i ≥1), CrV (critical value set for the design), and
The values of m1 and mi can be calculated as follows:
, where:
the average percentage of expanded plots for CrV of 1, and the average percentage of expanded plots for CrV of i.
This cost function will be greatly simplified by removing the last cost item from it when an alternative estimator without need for tree coordinate collection becomes available.
3.2.2 Plot design II
Following the same way as above, the cost function for plot design II with the estimator used in this study is developed according to the number of relevant tree search circles, which is 5 for an initial sample plot satisfying the condition to adapt and 13 otherwise, and the size of a relevant tree search circle, which is 9 times larger than the area of the initial sample plot as shown in Figure 3.7. It is presented as follows:
[ ] =
where:
total cost in terms of timefor plot design II,
average time requirement of the round travel between the camp and the inventory region, average time requirement to travel to next sample point,
average time requirement for field observation of the target variable per initial sample plot area,
average time requirement for collection of coordinates of relevant trees of the sampled trees per initial sample plot area,
59 sample size or number of initial sample plots,
expected value of the number of initital plots with the presence of at least one tree, expected value of the number of initial plots with the presence of at least i trees(i ≥1), and
CrV (critical value set for the plot design)
The calculation of m1 and mi can be done with the formula given for plot design I.
The cost for the travel between subplots and relevant tree search circles is not taken into account as it is normally very small in comparison with the cost for travels from sample point to sample point.
If an alternative estimator without need for tree coordinate collection comes into existence in the future, then the last cost item can be simple dropped.
3.2.3 Plot design III
The cost function composed for plot design III is:
where
total cost in terms of timefor plot design III,
average time requirement of the round travel between the camp and the inventory region, average time requirement to travel to next sample point,
average time requirement for field observation using initial BAF at a sample point,
time requirement for field observation using final BAF at a sample point where the condition to adapt is met,
average time requirement for collection of coordinates of relevant trees of the sampled trees at a sample point,
sample size or number of sample points,
expected value of the number of sample points with at least one tree tallied with an initial BAF,
60 expected value of the number of sample points with at least i (i≥1) trees tallied with an initial BAF, and
CrV (critical value set for the plot design)
The values of and can be calculated as follows:
, and where:
the average percentage of adapted sample points for CrV of 1, and the average percentage of adapted sample points for CrV of i.
Unlike in plot design I and II, in the last cost item of the cost function for plot design III cannot be fixed as the relevant tree search circle for a sampled tree has a variable size as mentioned before. This will bring some difficulty for the comparison of overall efficiency.
3.3 MATERIALS