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# What Type of Plot is "Most Forgiving?"

Over the years, practicing foresters build up many rules of thumb. One of the "rules" that we've heard is that variable radius plots are more "forgiving" than fixed radius plots when it comes to incorrectly including or excluding a borderline tree. Is this actually the case? Using a similar style of analysis from our previous Biometrics Bits columns, we can arrive at a data-driven answer.

VRP and FRP Sampling
The two types of plots we'll be considering today are variable radius plots (VRP) and fixed radius plots (FRP). Deepest apologies to you "clustered sausage sampling" fans out there - that analysis will have to wait for another day. (Yes, that is a real sampling method.)

A variable radius plot like a basal-area-factor 10 (BAF 10) plot uses a prism of a particular size to determine whether each tree is in or out of the plot. This means that large trees can be far from plot center and still be included, while small trees must be fairly close by to be included. VRPs are widely used in timberland assessment because they are generally faster than FRP plots to install. A VRP sampling method also makes it easy to calculate basal area (and thus volume) - simply tally the number of trees within the plot and multiply by the BAF of your prism. From a sampling theory perspective, VRPs also have the property that they minimize error by sampling in proportion to the variable of interest (volume).

Fixed radius plots like a 1/10 acre plot require a cruiser to measure all of the stems within a 37.2 foot radius. Because of the need to measure the diameter of each stem to obtain basal area, these plots can take a while to install. Tools like an ultrasonic DME can help reduce measurement time to make FRP more competitive with VRP.

However, the benefit of doing a FRP is that, since they comprise a fixed area, they tend to be more useful for remote-sensing work. The trouble with VRPs in remote-sensing is that big trees can sometimes be located far from the plot center, making it difficult to pair up the plot measurements with the imagery.

What is a "Forgiving" Plot Type?
Before we can dive into our analysis, we need to be specific about what we mean when we say a plot type is "forgiving." We can think about this in two ways. The first relates to the sensitivity of each plot type to mishandling borderline trees: how much can our basal area estimates change by erroneously including an ‘out’ tree in plot, or excluding one that should actually be in? The second relates to differences among cruisers, and whether someone is more likely to count trees as ‘in’ rather than ‘out.’ If a cruiser doesn't have a systematic bias in their treatment of borderline trees, there shouldn't be much of a difference between the estimates generated by different types of plots. But if a cruiser does have a particular "style" of handling borderline trees...

Our Cast of Cruisers
In theory, the correct way to determine whether a borderline tree is in or out is to measure the distance from the plot center to the center of the tree. If it is longer than the limiting distance, the tree is out.

In practice, many cruisers feel fairly confident in their ability to visually assess whether borderline trees are in or out. To examine the consequences of this belief, we analyzed the performance of three imaginary cruisers:
Pessimistic Pete tends to think borderline trees are out of the plot, calling 75% of borderline trees "out"
Optimistic Olivia tends to include borderline trees, calling 75% of borderline trees "in"
Even Evan takes a "fair and balanced" approach to borderline trees and calls 50% "in" and 50% "out"

We took a look at how Pete, Olivia, and Evan did at estimating basal area per acre for a stand given their tendencies. We also repeated this simulation many times to determine whether FRP or VRP showed more “spread” around their average estimates.

Simulating Cruises
As in some of our previous analyses, we used the US Forest Service's fantastic Forest Inventory and Analysis (FIA) dataset. We used a mixed hardwood / loblolly pine plot from Georgia as the "seed" for constructing a virtual stand that our cast of cruisers would sample. We used these data to generate a rectangular 200 acre stand with a basal area of about 75 ft, dominated by loblolly pine but carrying a significant hardwood component.

A map of the stems within the virtual stand

Within this stand, we randomly established 80 plot centers at least 100 ft apart and 75 ft from the stand bounds. For this analysis we compared 1/10th acre FRPs to BAF 10 VRPs.

The key aspect of this simulation was the treatment of borderline trees. We defined a borderline tree as one which was within +/-2 feet of its limiting distance from plot center. The various cruisers included or excluded borderline trees at their characteristic frequencies described above.

For each cruiser, we simulated 100 cruises with each plot type. We worked up the stand-level population estimate for each cruise and graphed them to evaluate the distribution of outcomes for each cruising style and plot type.

Results
Let’s take a look at how Pete (red), Olivia (green), and Evan (blue) performed in their FRP and VRP cruises (figure 2).

Distributions of 100 simulated cruises when including borderline trees with probabilities of 25% (red), 50% (blue), and 75% (green). The bold black line indicates the true basal area of stand (75 square ft/acre), and the dotted lines indicate the mean estimated basal area of each cruiser.

There are two things to pay attention to in this figure. First, not surprisingly, the cruiser who included borderline trees at the highest rate produced the largest basal area estimate, well above the observed basal area of the property. Conversely, including borderline trees with a probability of 25% resulted in an underestimation of basal area. Evan, who included every other borderline tree, was the closest (mean predicted basal area of about 73.5 square feet/acre). Second, note that for each cruiser the spread around their mean basal area estimate is as wide or wider when using VRPs rather than FRPs. This effect is particularly apparent for both the low (red) and high (green) probability cases, while the range of potential basal areas is about equal for the medium probability cruises (blue).

Increased sensitivity of VRPs can also be seen when estimating trees per acre (TPA). Comparing results when including trees with a 50% probability (figure 3), we see that VRP results in a wider range of possible TPA estimates across 100 cruises when compared with FRP. In the case of FRP, TPA varied from about 113-122 trees/acre, while the VRP cruise produced estimates spanning 115-130 trees/acre.

Trees per acre for 100 simulated cruises using FRPs (red) and VRPs (blue).

Discussion
So what can we take from these results? Well, for this particular stand at least, our analysis does not support the belief that VRPs are a more robust sampling method in regards to inclusion/exclusion of borderline trees. While different probabilities of including borderline trees affected the FRP and VRP cruises about equally on average, our results show that across many cruises the VRP plots tended to be as or more sensitive to error associated with handling borderline trees, particularly for trees per acre. In other words, even without changing their cruising approach, we expect a cruiser to produce a wider range of basal area and trees per acre estimates when using variable radius plots when compared to fixed radius plots.

So based on these results, would we recommend cruisers shy away from VRPs? Not quite. Recall that the least biased cruiser (Evan, who including borderline trees with a 50% probability) was quite accurate with both methods. However, even his VRP cruises showed as much sensitivity to borderline trees when compared to FRPs, and his ‘every other tree’ rule of thumb could have swung the BA estimate 5 square feet/acre in either direction with both approaches. The main takeaway here is that, regardless of how you cruise, relying on rules of thumb over precise measurements may expose your cruise to potentially costly errors such as those we see here with handling borderline trees. If a high degree of accuracy is required, you’ll need to take this into consideration regardless of how many plots you install.

As for our main question of whether VRPs are more “forgiving” than FRPs, our results suggest this isn’t the case. So, don’t fool yourself into thinking you’re protected from measurement error based on the sampling protocol you use. If you prefer to use FRPs, either for easy linking with remote sensing data or another reason, our results show that you aren’t sacrificing robustness in your plot design, and may in fact enjoy significant gains, especially for estimates of trees per acre.

This type of analysis provides a framework for addressing many further questions around measurement error. Do these trends in VRP vs. FRP robustness extend to other forest types? What about different "borderline tree strategies" - particularly those that are more careful with larger or smaller trees? How does the size of the borderline buffer influence the results? We look forward to exploring these questions in future articles.