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I am currently working on biomass estimates using satellite imagery. I'll quickly define the background of my question, and then explain the statistical question I am working on.

Background

Problem

I am trying to estimate biomass over an area in France. My response is the steamwood volume density (in $m^3/ha$), which is more or less proportional to biomass (depending on the wood densities...).

The independent variables that I have are vegetation indices derived from measured reflectances over this area (the satellite used in the study is MODIS for those who know it). These indices are for example NDVI, EVI, etc. I have maps of the indices, and the resolution of the maps are 250m.

There are strong correlations between these indices and the volume in a same forest type (biome and climate). So I am trying to regress the volume density against these indicators (actually their time series) on inventory plots where I know the volume.

Forest inventories

The volume on these plots is estimated with the following sampling method:

  1. Inventory nodes are placed on a regular grid covering the area.
  2. A plot is attached to each node, and the inventory process (tree types, volumes, canopy height, etc.) occurs on this plot. Of course I am interested only in the inventory plot and the values of my vegetation indices is the value of the pixel containing the plot.
  3. The inventory process on a plot is the following:

    http://i.stack.imgur.com/DeHdC.png

    • Measure of the trees that have a diameter > 37.5cm in the 15m radius circle
    • Measure of the trees that have a diameter > 22.5cm in the 9m radius circle
    • Measure of the trees that have a diameter > 7.5cm in the 6m radius circle

The volume density is then calculated using expansion factors.

For each plot I have access to the data for all the measured trees.

Moreover, for each single tree, I have an uncertainty on the volume due to the use of allometric equations (let us say 10%).

Where statistics are important...

For my regressions to be more accurate, I need for each estimate of volume the variance/CI of this measure. This depends, IMO, on the number of trees sampled and the volume density found.

So I have two problems:

  1. How to account for the fact that my vegetation indices are measured over a pixel of 250m ?

    I can assume that the volume density is constant over one pixel, and that I sample this pixel with one inventory plot.

  2. How to estimate the variability of my volume density?

    I think I could use bootstrapping on the tree population. But my total number of trees measured can be pretty small (from 7 to 20...). Moreover, how can I take into account the fact that I am measuring the trees on different circles depending on their sizes ? And how should the variability change if I am looking on an entire pixel ?

I was also thinking that I could use a Monte Carlo Simulation to simulate a forest, and then randomly sample this forest with plots to see what is going on...

I do not have a strong statistical background, so I am a little bit lost!

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I don't understand your data very well, but I can tell you that an alternative to the multinomial bootstrap that works better for rare events is perturbation / wild bootstrap. Perturbation is extremely flexible and is often able to handle non-iid data, however sometimes a great deal of finesse is needed to correctly approximate the cdf. If you succeed in correctly specifying the bootstrap formula, you will make fewer assumptions and likely be less biased than the smoothing method suggested previously, particularly given your sparse dataset, which may make density estimates unstable.

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If I had to approach this problem I would first start by:

  1. looking at a map of the source data
  2. attempting some sort of 2d smoothing on the surface, try to inform it with AIC
  3. compute the derivative of the smooth at the location and relate variation in input to variation in output using the delta method
  4. Compare the results of this to some "known" values in order to verify/validate the approach

Relevant links: http://www.stanford.edu/class/cme308/notes/TaylorAppDeltaMethod.pdf http://www.ingentaconnect.com/content/klu/stco/2010/00000020/00000004/00009140?crawler=true

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