OK, I had an interesting call the other day with someone who was inquiring about high accuracy lidar data for looking at beach renourishment volume calculations. They were talking about accuracies (RMSE) on the order of 3 to 5 cm – an inch or two. My first take was yeah, that is probably where things are going – paying for sand gets expensive and knowing exactly how much you are getting has its obvious advantages. We have worked on uncertainty for sea level/inundation mapping and it was pretty clear that the data can, in some locations, have a pretty dramatic effect on the mapped location of the inundation extent. I thought that the beach volume example may be a nice corollary to that work. I was surprised at the answer and found that a seemingly trivial oversight created far more problems (I will cover that in a later entry – this one got long, sorry).
So let’s assume we have a lidar data set with an RMSE of 10 cm in open terrain (pretty typical of many collections) and a mean of zero – i.e., the data is not biased (again, pretty typical of open terrain). Since we are dealing with a beach – the open terrain assumption is pretty valid – sure, a little dune vegetation at the landward extent, but not like a forest or anything. I chose a beach somewhere in New Jersey to highlight the example and a 1 km by 100 to 140 m ‘Area of Study’ along the shore to compute volume changes (Figure 1).
The area of study is 126,750 square meters and the average elevation is 1.782 meters. So, just for argument let’s assume the beach is flat and 1.782 meters high, the volume in each square meter would be 1.782 cubic meters and have a standard deviation (SD) of 0.1 cubic meters (10 cm x 1 m x 1 m). I inserted SD in there instead of RMSE since it is pretty universally agreed that in open terrain lidar data is normally distributed and, in this case, non-biased. There is a very small difference between SD and RMSE, depending on the sample size vs. population, but for this example, the difference is trivial.
Back to the example – if we add all the cubes together we get 225,868 cubic meters of sand above the 0 elevation (NAVD88). But what is the total error – how much more could there be or not be, and am I looking at the possibility of paying money for sand I already have on my beach?
So, 1 SD is going to cover about 68% of the data, but to be sure – let’s go to 95% – which is roughly 2 SD (2 SD = 95.4%). The volume in this case would be 1.782 +/- .2 cubic meters or about 1.782 +/- 11% cubic meters. Eleven percent is not a trivial value – especially when looking at 225,000+ cubic meters (24,750 cubic meters). That is a lot of truckloads of sand!
It does not look good at this point, but it gets a lot better thanks to the normal distribution (which has been a lifesaver to me in school). Remember, some values are above the mean and theoretically the same below the mean. So, instead of carrying the error straight up the chain, we know that each of the volume estimates has a chance to be wrong in the opposite direction and, in theory, would converge on the mean (Central Limit Theory).
To calculate the SD of the total of all cubes, we use a sum of squares calculation:
SD of total surface = (0.12 + 0.12 + ……)0.5
If you add (0.12) 126,750 times (oh yeah, multiplication works too) then take the square root – you will find that the SD of the surface is ………………………… 35.6 cubic meters – so at 95% confidence it is like 70 cubic meters (2 x 35.6). Seventy cubic meters is a bit better than 24,750!!!!
Suppose we are now looking at two surfaces – before and after a storm or a renourishment – which is probably more likely. Each surface has a SD – we calculated the first, but let’s say the second is from an older data set with an RMSE (SD) of 18.5 – that is still, and very importantly (!), not biased. It will have an SD of 66 cubic meters. We can, you guessed it, add those together using sum of squares and get 75 cubic meters. If we go to 95% – it is 150 cubic meters.
So, in theory we should be able to compare two surfaces at the location in Figure 1 before and after a renourishment or storm and be 95% certain to 150 cubic meters of the change (if the beach did not change between measurements!!). If we had the same data sets collected before and after a contractor said they put out 50,000 cubic meters and we measured it using our lidar at 51,000 – we could be very certain (and happy) that they were giving us about 850 cubic meters for free.
Which brings me back to the first sentence of this ‘story;’ looking at it again, I am not sure that higher accuracy lidar is necessary for beach renourishment/change measurements. The kicker, it turns out, is the bias – something I will discuss in the next ‘story’ about this New Jersey Beach – and to some degree, how the data is converted to a surface. Oh BTW, a product of the large numbers – the calculated value of the volume of sand above NAVD88 was 224,935 cubic meters (and depends on the analysis cell size) – the ‘flat beach’ estimate of 225,615 was not too bad – but outside of the 95% value!