# How do I calculate area error probability of a polygon with knowledge of point errors

I have a polygon, representing a glacier, which is defined by over 3000 points. This polygon is digitised from an accurate orthophoto. I also have a polygon of said glacier digitised from Landsat. In order to work out the error distribution of the points defining the Landsat polygon, I calculated the shortest distance of each Landsat-polygon point to the extremely accurate orthophoto-polygon. I now know the error distribution of points defining the Landsat polygon (normal distribution centred around a mean close to zero).

My question is: With knowledge of my point error distribution, how can I estimate an area error probability?

A long search has only shown me how to propagate point errors to area errors for simple shapes (such as squares) with knowledge of x and y errors of points.

I initially visited this site with this exact same issue and, while I'm not sure this will completely address the answer, this is what I came up with.

You might look at a 2012 M.S. Thesis by Sebastian Carisio from the University of Delaware entitled "Evaluating aerial errors in northern Cascade glacier inventories." You can find the manuscript here: http://www.morageology.com/view_pub.php?id=266

In the work I was doing (Final report can be found here: http://www.morageology.com/view_pub.php?id=12 ), this is how I addressed the error for glacial area (This is directly copied from the report I just linked):

Manually digitizing glacial boundaries inherently introduces error into final measurements. These errors include: 1) resolution and accuracy of the background image, 2) the accuracy and precision of placed vertices along glacial margins, 3) position uncertainty among debris- and non-debris-covered glaciers, and 4) natural variability of glaciers from year to year. Imagery errors depend on the source of the image. For example, Worldview 2 satellite images have a horizontal position error (CE90 – Circular Error, 90% percentile) of approximately 5 m (16 ft) (DigitalGlobe 2016), whereas NAIP images impose a standard that 95% of well-defined points tested fall within 6 m (20 ft) of true ground points (Eckert 2011). Horizontal accuracy of vertices placed along glacier margins are a function of the scale of the map created and can be quantified by using standards employed by the United States Geological Survey’s National Mapping Program standards (USGS 2017). Such standards state the following: “horizontal accuracy as 90% of all measurable points must be within 1/30th of an inch for maps at a scale of 1:20,000 or larger.” For the purposes of this study, the horizontal accuracy of a point at 1:1,000 scale is 0.847 m (2.778 ft) and 1:5000 scale is 4.233 m (13.889 ft) based off of the USGS nomenclature.

Total surface area error, E, is therefore expressed as a combination of total measurable uncertainty, E1, along with potential variability error, E2. Total measurable uncertainty, E1, is defined by Carisio (2012) as:

E1 = Sqrt(Ai) * (p + u) * Sqrt(2)

Where Ai is the surface area of the delineated glacier, p is the horizontal uncertainty of the source image, and u is the horizontal uncertainty of a placed point. Potential variability error, E2, can vary as little as 2% and as much as 5% for glaciers in question; generally, smaller glaciers have a higher variability error (Riedel and Larrabee 2016; Riedel et al. 2015; Sitts et al. 2010; Post et al. 1971). For the purposes of this study, and to account for all other potential errors, a relatively high 5% value was used for potential variability error:

E2 = 0.05 * Ai

Total surface area error, E, is therefore calculated as the following:

E = E1 + E2

I hope this helps you and others with this question. There might be a better way to do this type of analysis, but other researchers (e.g. Riedel and others) who do work in the North Cascades have used this method and when I used it for my work, the formulas above generated error estimates that roughly agreed with those other authors.

Cheers!