THREE-P SAMPLING: AN EFFICIENT SAMPLING TECHNIQUE FOR FOREST INVENTORY
James C. Space and Doyle Turman
Abstracts
Recent applications have demonstrated a variety of uses for 3P sampling.Three-P sampling offers a valuable alternative for sampling populations where each tree can be visited. 1 t has found increasing use in more elaborate mu lti-stage sampling schemes utilizing equal probability, 3P, and variable probability list sampling.
Three-P sampling can be used in any application where a relative volume or value can .be assigned to prospective samples.
Further advantage can be obtained from a variety of tree measurement methods and the STX computer program. Sample tree volumes may be estimated using conventional volume tables or equations. However, precision can be improved through the use of felled tree measurement or optical dendrometry. The STX pro- gram is a flexible and efficient tool for compiling inventories utilizing various sam- pl ing schemes and dendrometry methods, without the expense and delay of custom- written programs or hand calculations.
These techniques offer advantages for improved efficiency when incorporated in existing inventories or in designs for new ones.
Waldinventuren mit Aufnahmewahrscheinlichkeiten, die proportional zu Vorhersagen sind (3P)
Die 3P-Methode kann sehr vielseitig angewendet werden. Bei Gesamtheiten, in denen jeder Baum aufgesucht werden kann, bietet die 3P-Methode eine wertvolle Alternative. 1 n mehrstufigen Inventuren haben sich Listenstichproben mit gleicher (3P) oder variabler Wahrscheinlichkeit als nützlich erwiesen. Die 3P-Probenahme kann überall dort angewendet werden, wo den Einheiten ein relatives Volumen oder ein relativer Wert zugeordnet werden kann.
Verschiedene Methoden der Baummessung und das Computerprogramm STX können weitere Vorteile bieten. Das Volumen von Probebäumen kann mit konven- tionellen Volumentafeln oder Volumenfunktionen geschätzt werden. Systematische Fehler können durch die Vermessung liegender Bäume oder durch Verwendung op- tischer Messgeräte vermieden werden. Das ST X-Programm ist ein anpassungsfähiges und nützliches Hilfsmittel zur Auswertung von Waldinventuren, das bei verschiede- nen Probenahmeverfahren und Messmethoden angewandt werden kann; somit las- sen sich die Kosten für die Erstellung von speziellen Programmen oder für manuelle Auswertungen einsparen.
Diese Methoden können die Effizienz sowohl von bestehenden als auch von neu zu planenden Erhebungen verbessern.
The authors are Forest Management Systems Special ist, Cooperative Forestry Staff, and Forester, Timber Management Staff, USDA--Forest Service Washington, D. C., U.S.A.
Sampl ing with probabil ity proportional to prediction (3P), developed by L. R. Grosen- baugh ( 1965), is a relatively new techn ique for incorporating the efficiencies of variable probability sampling in forest inventories when only rough overall population characteristics are known. Where the populations can be, or have been, completely enumerated in advance, variable probability list sampling can be used (Hartley, 1966; Schreuder, et al., 1968, 1971;
Stage, 1971; Schreuder, 1975). The latter has the advantage of a possible increase in efficien- cy through exact specification of the number and distribution of samples in advance. How- ever, in many forest inventory situations where a relatively large number of samples are to be selected, advanced lists are usually unavailable, or their preparation prior to sample selec- tion is impractical or infeasible. 1 n these situations, 3P sampling offers a practical alternative with little difference in efficiency (Grosenbaugh, 1975).
The advantage of variable probability sampling in obtaining a low statistical sampling error with a very small number of samples has also encouraged the development of several other techniques for improving estimates of tree volume and quality (Grosenbaugh, 1964, 1973a). These include the measurement of the sample treeswith optical dendrometers, there- by eliminating possible volume table or equation bias (Grosenbaugh, 1963), and the use of primary units of measure (cubic volume, surface area, and length), permitting direct conver- sion to estimates of manufactured product's volume and value (Grosenbaugh, 1967a; Tur- man, 1970; Space, 1974a). These, and a number of other sampling and measurementtech- niques, have been incorporated into a powerful computer program called STX, which can process inventories having up to three sampling stages (Grosenbaugh, 1974).
lnventory of Small Tracts Using 3P Sampling
A very basic appl ication where 3P sampl ing can be used to great advantage is the inven- tory of small tracts of timber in which each tree will be visited (Mesavage, 1965, 1971; John- son, et al., 1967; Grosenbaugh, 1967b, 1968; Caballero, 1974). This is a common situation in ourcountrywhen it is necessary to mark individual trees for cutting within a tract selected for harvesting. While 3P sampling permits many types and kinds of estimators tobe used - such astreevolume or value - we, in our applications, have generally used gross tree volumes either weighted or unweighted by relative value differences.
The whole premise of 3P sampling is that some estimate of tree volume or value - even a poor one - is better than a mere count, and these estimates may be corrected by sampling a small number of accurately measured trees. Furthermore, since the selection of samples is in direct proportion to the volume or value they represent, the larger, more valuable and more variable trees will be sampled more frequently.
The advantage of 3P sarnpling is the ability to obtain a very low statistical sampling error with a very few samples. For exemple: one hundred trees or less are commonly needed to obtain a sampling error of ± 2 per cent, two times in thr'ee, regardless of the number of trees on the tract.
After choosing an appropriate sampling error, the number of 3P sample trees desired is determined using conventional random sampling equations with one exception: the expres- sion of the coefficient of variation for 3P sampling is the variation of the ratios of the indi- vidual measured tree volume to its predicted tree volume. Therefore, any reduction in the variation of these ratios reduces the number of samples. 1 n practice, estimates of volume for individual trees, and the resulting ratios, are not as poor as one might expect. Even a person with very little forestry experience can tel1 that one tree is larger than another and, having
some reference volume for comparison, approximately how much !arger. A more experi- enced forester or technician can easily make consistent estimates whose ratios develop a co- efficient of variation of 20 per cent or less. With training and practice the variation can be reduced to 15 per cent or less, thereby maximizing the efficiency of 3P sampling.
The selection of the sample trees can be done with a conventional set of random num- bers by discarding all random numbers !arger than the estimated total population volume di- vided by the desired number of samples, or by using a device, such as developed by Laasase- naho (1973), which samples by diameter breast height cumulation. In our country we have found that a custom-generated computer list of random numbers for the specific sampling application is the best and easiest method to use to avoid the introduction of sampling bias.
The computer program THRP (later superseeded by RN3P), developed by Grosenbaugh (1965), will generatesuch a list based on the number of trees tobe visited, the volume of the largest tree to be sampled, and the estimated total population volume divided by the num- ber of sample trees desired.
During the inventory, as each tree is visited and marked, an estimate of volume is made and recorded. The timber marker then compares the recorded volume estimate with the next number on the random number list. Should the estimate equal or exceed the paired random number, the tree is selected as a sample tobe measured.
Measurement of the selected sample trees can be by any method consistent with preci- sion and accuracy requirements for the inventory. Should volume tables be sufficient, the needed stem measurements are taken and recorded. When precise volume estimates are re- quired, the standing tree bole may be measured with an optical dendrometer or felled and measured. The last two methods have the advantage of accurately defining the form of the stem and classifying the tree into product, quality, or defect segments. Where the possibility exists for considerable unseen defect or stem breakage due to felling, Johnson and Hartman (1972) recommend that the sample trees be felled and bucked into logs before measuring.
Simple 3P inventories employing volume tables or equations can easily be compiled by hand. Space ( 1973) devised field and office forms which simplify the summation process and require only a basic background in mathematics.
When dendrometry has been used, it usually is more efficient to compile the data using computer processing. The STX computer program developed by Grosenbaugh ( 1967c, 1974) and modified by Space (1974b) can process data from a wide range of sampling designs and dendrometry methods. This program converts dendrometer and/or direct measurements of the sample trees to primary un its of measure (cubic volume, surface area, and length), projects unseen portion of the tree bole, and interpolates to specified top diameter limits. The popu- lation frequency for each sample tree is calculated and volumes are summed by stratum, spe- cies, products, and grade. lf conversion coefficients are supplied to the program, they are applied to the summaries of the primary units of measure to produce predictions of product or mill outturn in volume or value. Either U. S. or S. 1. (metric) units of measure are accept- able as input and the user can specify that output summaries be made in either of these un its as desired (Grosenbaugh, 1973b, 1974). The program will also subdivide the sample tree boles by interpolation, based on user-supplied product length and trim allowance criteria, to produce summaries by diameter group of Scribner, Doyle, International 1/4-inch, or cubic (Smalian Formula) log scale. Of course, other log rules may be substituted for these in the computer program if desired.
lnventory of Large Tracts Using 3P Sampling
Subsampling of Plot or Point Sample lnventories
lnventories of large tracts of timber in the United States and Canada usually involve the installation of fixed-radius or variable-radius plots, or clusters of plots, to meet specified sam- pl ing objectives. Aerial photos are often used for stratification prior to plot selection. For the
initial measurement of an inventory using 3P subsampling, each plot must be visited and the species and DBH of each tree recorded. On fixed-radius plots, an estimate is made of the vol- ume of each tree. On variable-radius plots, either merchantable or total height of each tree is estimated, and basal area times height is used to predict the tree's volume. The 3P sample trees are then selected, based on these estimates or predictions, using the random-number technique mentioned before. The sample trees are measured with a dendrometer and processed through the STX program for total volume estimates (Space, 1974b).
1 n a pilot test of the method, Steber and Space ( 1972) found that 406 variable-radius plots with a 3P subsample of 221 trees on 93 of the plots gave a sampling error of +7.1 per cent on a tract of 1,012,000 acres (405,000 hectares), of which 6.95 per cent was attributable to the first stage sample alone.
Bonner ( 1972) compared 3P sampling with fixed- and variable-radius plots in Canada and found it to be the most efficient sampling method based on the crew time and sample vari- ance obtained. Use of 3P subsampling on fixed-radius plots was found tobe more efficient than fixed- or variable-radius plots alone. No test was made of the efficiency of 3P subsam- pling of variable-radius plots.
Other Multi-stage Sampling Techniques
In most multi-stage sampling schemes, the first-stage sampling error generally establishes the minimum sampling error that can be achieved; additional sampling stages, while they may add only slightly to total sampling error, would serve to reduce combined sampling error on- ly under very unusual circumstances. Therefore, any technique which serves to reduce this first-stage sampling error at minimal cost in worth exploring.
A sampling scheme used in Alaska (Dippold, 1974) for re-inventorying a large timber sale, consisted of subdividing 121 cutting units into 768 equal volume areas. Subdivisions of the cutting units were made on aerial photographs in order to delinate areas estimated to contain approximately equal volumes equivalent to that of a fully-stocked 10-acre stand. A sample of 35 areas was randomly selected from a list and their boundaries were located and marked on the ground.
In the second stage, each of the selected areas was 3P-sampled an the sample trees were felled and measured to determine net volume.
The sum of the individual tree estimates provided an unadjusted gross volume for each se- lected equal volume area. These area-accumulated gross volume estimates were adjusted to net volume based on the felled measurement of 3P sample trees.
The short working season in Alaska coupled with the remote location of this timber sale area required that the most efficient use of field time be considered in the inventory design.
Helicopters were necessary to transport crews to and from work areas. The fieldwork on the inventory of the 12,074 acres (4,886 hectares) was completed in 17 days and provided total volume estimates with the designed sampling error limits of ±5 per cent.
Another technique to reduce first-stage sampling error to a minimum is to predict volumes on aerial photo plots and select a subsample of these for ground measurment using variable probability list sampling. n turn, these ground plots may be further subjected to 3P sampling
to select trees for dendrometry. Perhaps the ultimate extension of this technique is the use of space imagery as the first sampling stage (Langley, et al., 1969; Langley, 1971; Nicholas, et al., 1973).
A similar technique for establishing a large number of inexpensive first-stage samples, and thus reducing sampling error, was developed by Wiant (1974). Volumes (on fixed-radius plots) or the sum of heights (on variable-radius plots) are estimated at points throughout the forest, and a subsample of these plots is selected for measurement by 3P sampling. All trees or a 3P subsample of the trees of these plots are then measured and expanded for the volume estimates.
Combination with Other Techniques
Stratification can be used in combination with any of these techniques to further reduce sampling error. For successive forest inventories, sampling with partial replacement (Ware and Cunia, 1962; Bickford, et al., 1963; Cunia, 1965) may be employed to take advantage of the correlation between measurements of the same trees or plots.
Use in Continuous Forest lnventory
Because of the large initial investment for permanent plot establishment and associated periodic remeasurement expenses, continuous forest inventory (CFI) methods and tech- niques should periodically be analyzed to improve efficiency and to reduce costs. A CFI scheme must be durable enough to accept frequent changes and/or shifts in management ob- jectives or in merchantability and utilization standards and product definitions which may be caused by improvements in wood processing technology.
Three-P sampling, dendrometry, and the STX program are valuable tools which can offer some cost savings and improved efficiency to the inventory specialist working with CFI. Bas- ically, 3P sampling can usually be used to reduce the number of required measured samples when establishing a CFI, or can be used very efficiently on existing inventories to subsample at interim or required remeasurement times. 1 n either case, with a reduced number of samp- les, a greater investment can be made in whole stem measurement of the sample trees with optical dendrometers. These measurements, processed through the STX program, can be con- verted to primary units to measure. They, in turn, can be used with the current adjustment conversion coefficients to develop the required volumes and/or values.
Should the first-stage sample consist of randomly-selected plots, there is usually not a great difference in cost between the initial establishment of a conventional inventory and one incorporating 3P subsampling - although much more useful and accurate information is obtained through the use of dendrometry. When the inventory is remeasured, however, only the plots with 3P sample trees need tobe revisited, and only the 3P sample trees need tobe remeasured. Previously established inventories need not be abandoned to incorporate 3P sub- sampling with dendrometry. In fact, it is usually important that the existing inventory sys- tem be retained for comparison with previous inventories. We have found, without excep- tion, that any valid and adequate inventory system can be adapted to 3P subsampling, den- drometry, and STX processing. All that is usually necessary is to select the 3P sample trees and measure them.
1 n an evaluation of two-stage 3P sampling, using National Forest Survey data for south- west Alabama,Van Hooser (1972) found that a standard inventory which would have re- quired the remeasurement of 6,500 trees on 456 plots could have been reinventoried by the measurement of only 259 trees on 170 plots - with an increase in the sampling error of only 0.2 per cent.
1 n a field test of the method, Van Hooser ( 1973) selected a 3P subsample of 342 trees on 258 plots within a survey unit containing 1,056 plots with a total of 10,400 trees. As a check on accuracy, the 3P subsample was compiled and compared with the original inventory. The volume estimate produced by the 3P subsample differed from the volume estimate using all trees on all plots by only 0.06 per cent. This was weil within the calculated standard error of the 3P subsample. Since the subsample which had been selected appeared adequate and un- biased, the selected trees were remeasured. When compared with the best available current information, the reinventory appeared to give an accurate prediction of growth, mortality, and removals. lngrowth from trees smaller than 5 inches DBH could not be measured, since no trees smaller than this were measured on the original inventory. No attempt was made to determine ingrowth by a subsample of the original survey plots. Using two-man teams, the remeasurement took 12 team-weeks to complete, or slightly more than 11 per cent of the 106 team-weeks required for the original inventory.
Remeasurement using a 3P subsample of an original inventory can be repeated until so many of the trees are lost through cutting or mortality that the sampling error obtained by measuring the remaining trees would exceed the standard for the inventory. The inventory base must then be reestablished.This will depend on the rate of mortality, cutting and/or the rotation length established. However, between 15 and 25 years would appear tobe feasible, based on experience to date.
Other multistage sampling techniques incorporating 3P sampling may be used in CFI.
Wiant ( 1975) updated an inventory by using plot volumes from the last inventory to 3P-se- lect a subsample of plots for remeasurements. Estimated plot volumes from aerial photos may be similarly used.
Other Applications of 3P Sampling
The use of 3P sampling should be considered whenever it is possible to make a quick and cheap prediction of some major variable of interest, variable-probability sampling is desired, there is little difference in the cost of measurement between items with small and large pre- dictions, and the population cannot be completely enumerated in advance.
Sample log scaling using 3P sample selection was proposed and tested by Space ( 1969).
Johnson, et al. (1971) further tested the method in Oregon and developed a computer pro- gram for compiling thedata.Three-P sampling was found tobe very efficient when compared with measurement of all logs in instances where high variability in load size or value preclude simple random sampling of log loads. As few as 100 logs may need to be measured for a desired sampling error of ± 2 per cent.
Chehock and Walker ( 1975) extended the use of the 3P selection process to draw a sub- sample of logs in combination with sample weight scaling of log loads. They found this to be advantageous, not only in decreasing the time needed to measure sample loads but also in allowing more detailed examination and measurement of multiple products within tree length logs.
Beaufait, et al. (1974) used 3P subsampling on transects to inventory logging slash fuels before and after fire treatment.
lt may also be feasibl_e to use 3P sampling in measuring forage use or browse transects for range or wildlife management.
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