Bin-centred difference scheme

For a bin-centred difference scheme:

1. start at bin <math>n = \frac{n_{\text{rmax}}}{2} + 1</math>
   1. start with <math>\delta</math> = 1
   2. if <math>\delta</math> is even compute the second order structure function <math>D(n,\delta)</math> as the segment mean of the square of the velocity difference between the bins separated by distance <math>\delta r_0</math> centered around bin <math>n</math>:
      
      <math>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>\delta</math> is odd compute the second order structure function <math>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>\delta r_0</math> centered on the upper and lower extent of bin <math>n</math>:
      
      <math>\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>\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>\text{ceil}</math> and <math>\text{floor}</math> indicate the upper and lower integer value respectively, then
      
      <math>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>t</math> in the data segment yielding a velocity difference after the application of the Level 1 QC criteria
   4. increment <math>\delta</math> and repeat steps until <math>\delta = n_{\text{rmax}}</math>
2. increment <math>n</math> and repeat steps until <math>n + \frac{n_{\text{rmax}}}{2}</math> exceeds the bin number for which valid <math>v^\prime</math> are available

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

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