Bin-centred difference scheme

For a bin-centred difference scheme:

1. start at bin [math]\displaystyle{ n = \frac{n_{\text{rmax}}}{2} + 1 }[/math]
   1. start with [math]\displaystyle{ \delta }[/math] = 1
   2. if [math]\displaystyle{ \delta }[/math] is even compute the second order structure function [math]\displaystyle{ D(n,\delta) }[/math] as the segment mean of the square of the velocity difference between the bins separated by distance [math]\displaystyle{ \delta r_0 }[/math] centered around bin [math]\displaystyle{ n }[/math]:
      
      [math]\displaystyle{ D(n, \delta) = \Big\langle \big[v^\prime(n+\frac{\delta}{2},\ t) - v^\prime(n-\frac{\delta}{2},\ t)\big]^2 \Big\rangle }[/math]
      
      where the angled brackets indicate the mean across all t for the data segment yielding a velocity difference after the application of the Level 1 QC criteria
   3. if [math]\displaystyle{ \delta }[/math] is odd compute the second order structure function [math]\displaystyle{ D(n,\delta) }[/math] as the segment mean of the mean of the square of the velocity difference between the bins separated by distance [math]\displaystyle{ \delta r_0 }[/math] centered on the upper and lower extent of bin [math]\displaystyle{ n }[/math]:
      
      [math]\displaystyle{ \Delta_{\text{lo}}(n, \delta, t) = v^\prime(n+\text{floor}\left(\frac{\delta}{2}\right),\ t) - v^\prime(n-\text{ceil}\left(\frac{\delta}{2}\right),\ t) }[/math]
      [math]\displaystyle{ \Delta_{\text{hi}}(n, \delta, t) = v^\prime(n+\text{ceil}\left(\frac{\delta}{2}\right),\ t) - v^\prime(n-\text{floor}\left(\frac{\delta}{2}\right),\ t) }[/math]
      
      where [math]\displaystyle{ \text{ceil} }[/math] and [math]\displaystyle{ \text{floor} }[/math] indicate the upper and lower integer value respectively, then
      
      [math]\displaystyle{ D(n, \delta) = \Bigg\langle \frac{\Delta_{\text{lo}}(n, \delta, t)^2 + \Delta_{\text{hi}}(n, \delta, t)^2}{2} \Bigg\rangle }[/math]
      
      the angled brackets again indicating the mean across all [math]\displaystyle{ t }[/math] in the data segment yielding a velocity difference after the application of the Level 1 QC criteria
   4. increment [math]\displaystyle{ \delta }[/math] and repeat steps until [math]\displaystyle{ \delta = n_{\text{rmax}} }[/math]
2. increment [math]\displaystyle{ n }[/math] and repeat steps until [math]\displaystyle{ n + \frac{n_{\text{rmax}}}{2} }[/math] exceeds the bin number for which valid [math]\displaystyle{ v^\prime }[/math] are available

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