VRP Sampling - Small Diameter Struggles

This article first appeared in the January issue of the Forestry Source.

Variable-Radius Plot (VRP) sampling has become the dominant method of sampling in forest inventory, but this hasn't always been the case. In this column, we'll look at how this elegant sampling method gained acceptance across the US, explore some common use and misuse cases, and examine a few properties of the method that should be considered when small diameter stems matter most.

The concept behind VRP sampling was first introduced in 1948 by Walter Bitterlich in Austria and picked up in 1952 by Lewis Grosenbaugh, who shared it with US foresters. Since then there have been a multitude of adaptations, tools, and teaching methods developed. In fact John Bell and Lu Alexander began teaching a short course on the then “new” method in 1957 which is still taught (John only just retired from teaching the course this year) and has benefited more than 2,000 cruisers[1] over the years. While the concept began 70 years ago as a new and controversial technology, VRP sampling has since become a tried-and-true mainstay of forest sampling.

Many of us first came in contact with the idea of VRP sampling in our undergrad mensuration class. Depending on your region and your professor, you likely know this sampling method by one of its many names:

  • Bitterlich sampling
  • Point sampling
  • Plotless cruising
  • Variable-plot sampling
  • Angle sampling
  • Prism cruising

One of the great things about VRP sampling is that it is easy to learn and fast to do. After choosing an appropriate basal-area factor (BAF - more on this later), you stand at plot center, swing around in a circle, and count the number of stems that appear larger than the slot in your angle gauge or prism. With the right choice of BAF, there are usually a small number of stems per plot and you can hammer out many plots in a single day.

When using a VRP sampling strategy, a general rule of thumb for determining the most appropriate BAF is:

-> BAF to Use = Approximate BA / Desired Tree Count <-

Your desired tree count will depend on the cost of cruising and the importance of precision of the resulting inventory estimates. There isn't a simple answer here, but using the cost + loss framework that discussed in previous Biometrics Bits articles can help you make a quantitatively-based decision about the best sampling method to employ.

VRP sampling has rightfully earned its place in a forester's inventory toolbox. But with such success comes a risk of misuse: the danger of being applied in every situation regardless of whether or not it is the best tool for the job.

To understand the strengths and weaknesses of VRP sampling, let's first develop an intuition for why it works. We've heard three main ways of thinking about VRP:

  • The first is to imagine a wedge of a huge circle with the point of the wedge located at plot center. As the wedge spins around, any tree that meets or exceeds the bounds of the wedge is "in" the plot sample. Larger basal area factors (BAF) mean larger angles for the wedge. This intuition maps directly on to how you actually measure a plot with a prism or angle gauge, but it's tough to see why the sampling math works.
  • The second way is to imagine that there is a circular "area of inclusion" around each tree. Larger trees have a larger area of inclusion. Larger BAFs generate smaller areas of inclusion (trees have to be bigger to be "in"). Any tree with an area of inclusion that overlaps with the plot center is "in" the sample.
  • The third way, and the one that makes the most intuitive sense to me, is to envision progressively larger concentric circles around plot center. Larger diameter classes have larger circles. Larger BAFs generate relatively smaller circles. If a tree falls within its respective diameter inclusion circle, it is "in." This way of thinking about VRP sampling is the closest to how conventional fixed-area circular plots work. By calculating the area of each of the concentric circles, it is straightforward to determine the number of trees-per-acre that each "in" tree represents (it is 1 / area_of_circle).

Armed with this intuition, we can understand the VRP issue we raised in an earlier Biometrics Bits article. In the June 2017 “Optimal Cruising For Your Forest Type” article, we mentioned that VRP sampling isn't always the best tool for the job and showed an example where we were surprised to find that fixed radius plots appeared to be a better solution. Many foresters who use VRP sampling reached out to us for a more detailed explanation.

In the original article, we had described a method for putting a dollar value on inventory precision by using a "cost + loss" framework. Most foresters know how much inventory data costs, but it is much harder to know how much money is lost due to imprecise information driving suboptimal management. The wider the confidence interval on an inventory estimate (the less precise it is), the more likely it is that non-optimal silvicultural decisions will be made.

Compared to common fixed-area sampling methods, common methods of VRP sampling result in higher imprecision on small stem stocking. To see why, think about the "concentric circles" intuition described above. Small stems are only counted if they fall within a small circle around plot center. Because the area of the circle is so small, each observation of a small tree represents many (sometimes hundreds!) trees per acre as shown in the table below. Typically very few small diameter stems are observed because the inclusion area is so small in commonly used BAFs relative to commonly used fixed-area plot methods. The combination of a small number of sporadic observations and the large expansion to trees-per-acre results in highly variable estimates for small diameter stems. This makes some common implementations of VRP sampling less attractive in younger stands where smaller diameter trees are driving the silvicultural decisions and future value of the stand.


As an example of this we have simulated a cruise in a mixed southeastern hardwood stand where 25 plot centers were established and various plot types were used to sample at the same locations and calculate the standard error percentage of the lower merchantable diameter range 6”-10”. You can see in Figure 1 that the standard error percentage is relatively higher in the lower diameter range for the VRP plots, especially for the larger BAFs. The difference between standard error percent drops as you move into higher diameters.


On the other hand, VRP sampling shines when precision for total volume or volume in larger diameter classes is the focus of the inventory. In those situations, it is a far more efficient method because small diameter stems matter very little and imprecision around those estimates is inconsequential.

To sum up, VRP sampling is an elegant and efficient tool for many forest types, but not all. For stands that require greatest precision in low diameters, you may want to reconsider using VRP sampling. In particular, think twice about VRP or the BAF selected when considering inventory methods for:

  • Stands to be thinned from above to capitalize on advanced regeneration
  • Planning for pre-commercial thinning operations

  1. http://www.john-bell-associates.com/ Note from Kim Iles. ↩︎