Bin-centred difference scheme: Difference between revisions

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For a '''bin-centred difference''' scheme:
For a '''bin-centred difference''' scheme:
# start at bin n = (n<sub>max</sub> / 2) + 1
# start at bin <math>n = \frac{n_{\text{rmax}}}{2} + 1</math>
## start with <math>\delta</math> = 1
## start with <math>\delta</math> = 1
## if <math>\delta</math> is '''''even''''' compute the second order structure function D(n,<math>\delta</math>) as the segment mean of the square of the velocity difference between the bins separated by distance <math>\delta</math>r<sub>0</sub> centered around bin n: <br /><br />D(n, <math>\delta</math>) = <math>\langle</math> [v’(n+(<math>\delta</math>/2), t) - v’(n-(<math>\delta</math>/2), t)]<sup>2</sup> <math>\rangle</math> <br/><br /> 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
## 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>: <br /><br /><math>D(n, \delta) = \Big\langle \big[b^\prime(n+\frac{\delta}{2},\ t) - b^\prime(n-\frac{\delta}{2},\ t)\big]^2 \Big\rangle</math> <br/><br /> 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
## if <math>\delta</math> is '''''odd''''' compute the second order structure function D(n,<math>\delta</math>) as the segment mean of the mean of the square of the velocity difference between the bins separated by distance <math>\delta</math>r<sub>0</sub> centered on the upper and lower extent of bin n: <br/><br /> dv'<sub>lo</sub>(n, <math>\delta</math>, t) = v’(n+floor(<math>\delta</math>/2), t) - v’(n-ceil(<math>\delta</math>/2), t) <br/> dv'<sub>hi</sub>(n, <math>\delta</math>, t) = v’(n+ceil(<math>\delta</math>/2), t) - v’(n-floor(<math>\delta</math>/2), t) <br/><br /> where ''ceil'' and ''floor'' indicate the upper and lower integer value respectively, then <br/><br /> D(n, <math>\delta</math>) = <math>\langle</math> [dv'<sub>lo</sub>(n, <math>\delta</math>, t)<sup>2</sup> + dv'<sub>hi</sub>(n, <math>\delta</math>, t)<sup>2</sup>] / 2 <math>\rangle</math> <br/><br /> 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
## 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>: <br/><br /> <math>\Delta_{\text{lo}}(n, \delta, t) = b^\prime(n+\text{floor}\left(\frac{\delta}{2}\right),\ t) - b^\prime(n-\text{ceil}\left(\frac{\delta}{2}\right),\ t)</math> <br/> <math>\Delta_{\text{hi}}(n, \delta, t) = b^\prime(n+\text{ceil}\left(\frac{\delta}{2}\right),\ t) - b^\prime(n-\text{floor}\left(\frac{\delta}{2}\right),\ t)</math> <br/><br /> where <math>\text{ceil}</math> and <math>\text{floor}</math> indicate the upper and lower integer value respectively, then <br/><br /> <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> <br/><br /> 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
## increment <math>\delta</math> and repeat steps until <math>\delta</math> = n<sub>max</sub>
## increment <math>\delta</math> and repeat steps until <math>\delta = n_{\text{rmax}}</math>
# increment n and repeat steps until n + (n<sub>max</sub> / 2) exceeds the bin number for which valid v’ are available
# increment <math>n</math> and repeat steps until <math>n + \frac{n_{\text{rmax}}}{2}</math> exceeds the bin number for which valid <math>b^\prime</math> are available
 
See [[Example bin-centred difference | example bin-centred difference calculation]] for more detail regarding the calculation
 
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Return to [[Processing your ADCP data using structure function techniques | Compute structure functions and dissipation estimates]]
 
[[Category:Velocity profilers]]

Latest revision as of 12:56, 23 May 2022

For a bin-centred difference scheme:

  1. start at bin n=nrmax2+1
    1. start with δ = 1
    2. if δ is even compute the second order structure function D(n,δ) as the segment mean of the square of the velocity difference between the bins separated by distance δr0 centered around bin n:

      D(n,δ)=⟨[b′(n+δ2, t)−b′(n−δ2, 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 δ is odd compute the second order structure function D(n,δ) as the segment mean of the mean of the square of the velocity difference between the bins separated by distance δr0 centered on the upper and lower extent of bin n:

      Δlo(n,δ,t)=b′(n+floor(δ2), t)−b′(n−ceil(δ2), t)
      Δhi(n,δ,t)=b′(n+ceil(δ2), t)−b′(n−floor(δ2), t)

      where ceil and floor indicate the upper and lower integer value respectively, then

      D(n,δ)=⟨Δlo(n,δ,t)2+Δhi(n,δ,t)22⟩

      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 δ and repeat steps until δ=nrmax
  2. increment n and repeat steps until n+nrmax2 exceeds the bin number for which valid b′ are available

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

Return to Compute structure functions and dissipation estimates

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