Bin-centred difference scheme

For a bin-centred difference scheme:

1. start at bin $n=\frac{n_{\text{rmax}}}{2}+1$
   1. start with $\delta$ = 1
   2. if $\delta$ is even compute the second order structure function $D(n,\delta )$ as the segment mean of the square of the velocity difference between the bins separated by distance $\delta r_0$ centered around bin $n$:
      
      $D(n,\delta )=⟨[v^{\prime }(n+\frac{\delta }{2},t)-v^{\prime }(n-\frac{\delta }{2},t){]}^2⟩$
      
      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 $\delta$ is odd compute the second order structure function $D(n,\delta )$ as the segment mean of the mean of the square of the velocity difference between the bins separated by distance $\delta r_0$ centered on the upper and lower extent of bin $n$:
      
      ${\mathrm{\Delta }}_{\text{lo}}(n,\delta ,t)=v^{\prime }(n+\text{floor}\left(\frac{\delta }{2}\right),t)-v^{\prime }(n-\text{ceil}\left(\frac{\delta }{2}\right),t)$
      ${\mathrm{\Delta }}_{\text{hi}}(n,\delta ,t)=v^{\prime }(n+\text{ceil}\left(\frac{\delta }{2}\right),t)-v^{\prime }(n-\text{floor}\left(\frac{\delta }{2}\right),t)$
      
      where $\text{ceil}$ and $\text{floor}$ indicate the upper and lower integer value respectively, then
      
      $D(n,\delta )=⟨\frac{{\mathrm{\Delta }}_{\text{lo}}(n,\delta ,t{)}^2+{\mathrm{\Delta }}_{\text{hi}}(n,\delta ,t{)}^2}{2}⟩$
      
      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 $\delta$ and repeat steps until $\delta =n_{\text{rmax}}$
2. increment $n$ and repeat steps until $n+\frac{n_{\text{rmax}}}{2}$ exceeds the bin number for which valid $v^{\prime }$ are available

See example bin-centred difference calculation for more detail regarding the calculation

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