Accumulating Errors in Beach Volumes: How to Estimate Error Bars?

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In summary: The problem is that the deltas are not independent, they are coupled by the data.In summary, the conversation revolved around the use of GPS nearshore bathymetry and beach topography data for estimating sand volumes on beaches. The speaker had interpolated the data using two different methods, objective mapping and a scale controlled linear smoother, and was concerned about the errors that may arise in estimating sand volumes. They mentioned the sources of error, such as measurement bias and noise, and were looking for ways to estimate these errors accurately. The speaker also expressed the need to be cautious when interpreting the data and identifying trends in sand erosion and accretion. They were seeking guidance on how to account for different types of errors and how to accurately estimate them
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curious_ocean
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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!
 
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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)
 

Related to Accumulating Errors in Beach Volumes: How to Estimate Error Bars?

What is the purpose of estimating error bars in beach volumes?

The purpose of estimating error bars in beach volumes is to determine the uncertainty or variability in the measurements of beach volumes. This allows for a more accurate representation of the data and helps to determine the reliability of the results.

What factors contribute to the accumulation of errors in beach volume measurements?

There are several factors that can contribute to the accumulation of errors in beach volume measurements. These include human error, instrument error, environmental conditions, and the complexity of the beach morphology.

How can error bars be estimated in beach volumes?

Error bars in beach volumes can be estimated through statistical methods, such as standard deviation or confidence intervals. These methods take into account the variability of the data and provide a range of values that represent the potential error in the measurement.

Why is it important to consider error bars in beach volume measurements?

Considering error bars in beach volume measurements is important because it helps to determine the level of uncertainty in the data. This can affect the interpretation of the results and the overall accuracy of the study. It also allows for comparisons between different studies and locations.

How can errors in beach volume measurements be minimized?

To minimize errors in beach volume measurements, it is important to follow standardized protocols and use reliable instruments. Data should also be collected multiple times and at different locations to account for variability. It is also important to carefully analyze and interpret the data, taking into account any potential sources of error.

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