# Calculating basal area per acre from variable radius plots

I have a variable-radius forest inventory that contains diameter at breast height (DBH) and basal area factor (BAF) data for each tree. I'd like to calculate the basal area per acre, but since the plots were collected with a variable radius, I don't have an area estimate. I know multiplying the "in" trees from a prism by the BAF can produce an estimate of the basal area per acre, but I don't know if the number of trees in each plot of the data corresponds to an actual count of "in" trees when they were measured in the field. That is, using this method, is the convention to record the DBH of all "in" trees, in which case I could simply count the trees and multiply by the BAF? Or, is there an alternative way to calculate basal area per acre using the DBH in combination with the BAF?

According to resources like the Alabama A&M Extension, the convention is to measure the DBH of all the "in" trees, so you could just multiply the count by the BAF to get an estimate of basal area. However, I'd recommend making sure that your data was collected following convention. Do you have access to a metadata file or to the methods used to inventory these plots? If so, refer to those.

Edit: Why can we just multiply the number of "in" trees by the BAF, and not worry about the DBH?

This is because the DBH of the trees is already implicitly considered when using a basal area prism to determine which trees are "in." The DBH of the trees in the basal area plot is just collected for getting additional estimates, like standing volume.

Why does the number of "in" trees give us an estimate of basal area? And how is DBH implicitly included in this estimate? It's not really intuitive, but I'll try to paraphrase an explanation from this technical bulletin:

Imagine you randomly choose a point in a forest. This point has a chance of being inside of a tree. Your chance of being inside of a tree is equal to the basal area of the forest in units of acres/acres. For example, if you are in a forest with a basal area of 80 ft^2/ac, the basal area is equal to 0.00184 ac/ac, so you have a 0.184% chance of choosing a point in a tree. If you count the number of trees you are inside (either 1 or 0), on average, you will count 0.00184 trees. If you multiply the number of trees you are inside (either 1 or 0) by a BAF of 43,560 (the number of square feet in one acre), on average, you'll get the basal area of the forest. Of course, this would be a horrible way to estimate the basal area of the forest, because even though on average you'd get the right number, you'd actually get either 0 or 43,560 ft^2/ac each time you measured.

Instead, suppose you draw imaginary circles around each tree that are 10 times as wide as the tree, and therefore have 100 times the basal area of the tree. Now choose a random point and count the number of circles you're in. Just like how, in the last example, the number of trees your point is inside is an estimate of the basal area of the forest, the number of imaginary circles your point is inside is an estimate of 100 times the basal area of the forest (in ac/ac). Divide the number of circles you are inside by 100 and multiply by 43,560 (i.e., multiply by a BAF of 435.6) to get the basal area in ft^2/ac. This gives a much more precise estimate now, since randomness won't have as much of an effect. And it takes into account the DBH of the trees, since the diameters of the imaginary circles are proportional to the diameters of the trees.

This is how basal area prisms work. They let you know whether you're inside the imaginary circle for each tree, geometrically taking into account the DBH and your distance from the tree.

Imaginary circles around trees. From the Wisconsin Conservation Department Technical Bulletin no. 23.

• I guess what confuses me about this is why the DBH wouldn't be used Apr 15 at 21:52
• @brlockwood Great point. I edited my answer to try to explain why. Apr 16 at 3:24