# Accumulation of errors

1. Jun 4, 2015

### curious_ocean

I've got GPS nearshore bathymetry/beach topography sand elevation data from the backbeach out to 8 meters depth, with 1 m spacing in the cross-shore, 100m spacing in the alongshore, and stretching multiple kilometers of coast. I have interpolated the data using a couple different interpolation schemes. We have many many elevation surveys over time (~300 or so), but at the moment I am only interpolating each survey in space. One interpolation method I use is objective mapping (despite the fact that my data set violates many of the assumptions that objective mapping requires - beaches don't have nice statistics!)
ftp://brigus.physics.mun.ca/pub/zedel/P6316/2011/bretherton_davis_fandry.pdf
(objective mapping is similar to kriging)
and the other interpolation method is a scale controlled linear smoother
http://www.sciencedirect.com/science/article/pii/S0025322702004978
Both of these interpolation methods give me an estimate of errors due to the interpolation and GPS RMS errors. We also have GPS bias that we need to account for.

Ultimately I want to use the interpolation maps to estimate a time series of sand volumes on my beaches. The problem is that the errors add up pretty quick in these volume estimates and I have to be really careful when interpreting these curves! I need to do as best I can to estimate the error bars on my volume time series so that I can figure out what is signal and what is noise! (ps- It looks like there are some really interesting long term erosion and accretion trends in my data set. I have many ideas I want to try in order to figure out where the sand is going and come from!)

My question is how do the error estimates from the elevation interpolations add up in a volume estimate? My guess is that a simple sum might overestimate the errors. I also anticipate that I will need to treat the interpolation/measurement RMS errors differently than the measurement bias?

(Real data is not pretty so I will just need to pick the best possible method even though I'm certain it won't describe my data perfectly!) Thanks in advance for the help!

2. Jun 4, 2015

### DEvens

You need to find a way to estimate the error based on the nature of your data. In general this is a non-trivial question.

You are already aware there are different sources of error. For example, one source is bias as you have identified. Estimating bias is often very difficult. You need some sort of "authoritative" data to compare with. Maybe you can compare to some area of sand that was accurately measured some other way? I don't know what that might be. Maybe somebody with a surveyor's equipment? Pretty tedious out to a depth of 8 meters.

Another is measurement noise. If you were to apply the same exact measurement scheme two times with only a short interval between, such that the sand should not have changed significantly, then you will get different values. This is measurement error. It can be physical (maybe water waves are confusing the sensor) or it can be simple accuracy limits (half the smallest measurement grid sort of thing). Hopefully such noise does not tend to push the data preferentially in one direction. So the average should be close to the true value. If that is the case there are a variety of ways to estimate the effects. For example, look at the bootstrap method, and related methods.

http://en.wikipedia.org/wiki/Resampling_(statistics)

If you know the formulas that produce the results you can do analytic estimates. There are a variety of ways to approach this. Some "buzz words" to look for are Latin squares and Monte Carlo. Basically what you are looking for is something along the following lines. (This is massively simplified.)

val = f(x, y, z)

delta val = f(x + delta x, y+ delta y, z + delta z) - f(x,y,z)