Estimation 1 - Error analysis

We have observed two very notable under-estimations:

1. The width of the visible wall and therefore the effective width of the building \(w\)
2. The length of the building \(l\) Since both factor into the calculation of the area of the rooftop our error is magnified quadratically.

First we consider the error in the estimation of \(w\). We assumed in Rooftop solar estimate that the blocks covering the outer wall are quadratic in shape and hence allow us to estimate the width of the building from the height of a single floor.

We have estimated the height of a storey by looking at the free space within a flat and arrived at a height of 3 metres. We have neglected the thickness of the actual floor. To estimate the thickness of the floor we start, again with a lower bound. Here 10 cm seems like an extreme lower bound. A concrete floor the width of an E-reader seems almost unreasonably thin. On the other end the floor is likely thinner than 1.5 m. Then our estimate will be

$$h = \sqrt{0.1 \cdot 1.5} = \sqrt{0.15} \approx 0.39 [m]\rightarrow 40 [cm]$$

Using this we correct our estimate \(h_{floor} = 3 \rightarrow 3.4[m]\) and therefore the observable width of the building to \(w=4\cdot 3.4 [m] = 13.6 [m]\). The width of the visible wall would then be \(3\cdot 3.4 [m]= 10.2 [m]\) which appears to still severely underestimate the distance between corners 1 & 5. Here we note that the floor sketch encompasses the outer perimeter of the building and thus also the width of the balcony. A balcony, however, has no roof and should thus not be counted. From a glance we can estimate that the balcony floor is somewhere between 1 and 2 m and we would thus have to subtract roughly \(\sqrt{2}\approx 1.41 [m]\) from our verification estimate, bringing the reference distance between corners 1 & 5 down to \(10.8 [m]\). This is closer to our estimate!

Similarly, the building length has been estimated using its width, propagating the estimation error and the reference distance between corners 1 and 2 has included the width of the balcony. After correction we get \(\sqrt{3\cdot 5}\cdot 10.2[m]\approx 39.5[m]\) and \(43.4 [m]\).

With the improved estimates we then obtain an effective roof area of \(39.5\times 13.6 \approx 537 [m^2]\) and a reference area of \((15,5-1,4)\times(44,08-1,41)\approx 601 [m^2]\) resulting in an approximation error of \(601/537 - 1 \approx 11 \%\). Close enough to 10% that we may be satisfied with the estimate.