Here I want to calculate a range of fractional BOE area values and display them over various common Arctic Ocean graphics in plausible areas and shapes for remnant ice. For example, what does two million sq km of ice look like up against the Canadian coast?

According to wikipedia, a polar cap area on a spherical earth (not quite our WGS84 ellipsoid) can be calculated by a simple formula given the radius of the earth and the latitude of the bottom of the cap. Having some target areas for BOE in mind, inverting the area formula will give the cap latitude yielding them.

The formula given is area = 2 pi r*r (1-cos (theta)) where theta is the polar angle included from the center of earth to the pole and to the bottom of the cap. This is not the latitude but rather its complement lat = 90-theta. Solving for theta for target area with the shorter polar radius of the earth taken as 6357 km amounts to finding a spreadsheet to provide the arccos function and convert radians to degrees:

90 - theta = 90- degrees(acos(1-(target area ÷ 2 *3.1416 * 6357 *6357)))

https://en.wikipedia.org/wiki/Spherical_captarget area cap lat

500000 86.40

1000000 84.91

1500000 83.77

2000000 82.80

2500000 81.95

3000000 81.18

3500000 80.48

4000000 79.82

4500000 79.20

5000000 78.61

These need to be checked against a daily NSIDC matching area during late melt season (eg four million sq km but Greenland Sea and CAA unwelcome minor contributions) because the polar stereographic projection is not equal area -- the pole region is off by 5-6%. However for simply visualizing what remnant ice would look like, it is good enough.

The outcome shows the 85º latitude is convenient for BOE 1.0 as it comes provided on products such as AMSR1_UHH. The 80th parallel is similarly appropriate to the four million BOE 4.0. Elsewhere, the radius for the circle tool centered on the pole can be figured after pixels/degree are calculated from parallels provided or from known land latitudes such as the Bering Strait at 65.89°.

After drawing the polar cap out to the chosen radius in an auxiliary layer and filling it, the pixel count should be noted in the histogram tool. This will vary depending on the scale of the underlying image resource, eg OsiSaf, AMSR2_large, Mercator Ocean etc etc.

Next, the move and loop tools in Gimp freeware can be used to move the cap area to another location and to another shape while holding the pixel count at the histogram constant. For example, the fractional BOE area might adjusted so its bottom uses the upper contours of the CAA islands.

Some examples in a bit. The final shape favored will vary from person to person for each fractional BOE as the 2012-2020 shapes of remnant ice vary quite a bit.