Bin-centred difference scheme

For a bin-centred difference scheme:

1. start at bin n = (n_max / 2) + 1
   1. start with [math]\displaystyle{ \delta }[/math] = 1
   2. if [math]\displaystyle{ \delta }[/math] is even compute the second order structure function D(n,[math]\displaystyle{ \delta }[/math]) as the segment mean of the square of the velocity difference between the bins separated by distance [math]\displaystyle{ \delta }[/math]r₀ centered around bin n:
      
      D(n, [math]\displaystyle{ \delta }[/math]) = [math]\displaystyle{ \langle }[/math] [v’(n+([math]\displaystyle{ \delta }[/math]/2), t) - v’(n-([math]\displaystyle{ \delta }[/math]/2), t)]² [math]\displaystyle{ \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 D(n,[math]\displaystyle{ \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 }[/math]r₀ centered on the upper and lower extent of bin n:
      
      dv'_lo(n, [math]\displaystyle{ \delta }[/math], t) = v’(n+floor([math]\displaystyle{ \delta }[/math]/2), t) - v’(n-ceil([math]\displaystyle{ \delta }[/math]/2), t)
      dv'_hi(n, [math]\displaystyle{ \delta }[/math], t) = v’(n+ceil([math]\displaystyle{ \delta }[/math]/2), t) - v’(n-floor([math]\displaystyle{ \delta }[/math]/2), t)
      
      where ceil and floor indicate the upper and lower integer value respectively, then
      
      D(n, [math]\displaystyle{ \delta }[/math]) = [math]\displaystyle{ \langle }[/math] [dv'_lo(n, [math]\displaystyle{ \delta }[/math], t)² + dv'_hi(n, [math]\displaystyle{ \delta }[/math], t)²] / 2 [math]\displaystyle{ \rangle }[/math]
      
      the angled brackets again indicating the mean across all t 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 }[/math] = n_max
2. increment n and repeat steps until n + (n_max / 2) exceeds the bin number for which valid v’ are available