Processing your ADCP data using structure function techniques

1. Extract or compute the along-beam bin center separation [r0] based on the instrument geometry
2. Calculate the along-beam velocity fluctuation time-series in each bin, [v’(n, t)] from the Level 1 along-beam velocity data that has met the QC criteria
   1. If using burst sampling, calculations are done over the length of the burst or some sub-period over which the turbulent flow statistics can assumed to be stationary
   2. If using continuous sampling, calculations are dome over segments with a duration over which the turbulent flow statistics can assumed to be stationary
   3. For each data segment consisting of N profiles, the turbulent fluctuations are calculated separately for each beam and bin around either:
      • The mean over the data segment
      • A linear detrend of the segment
      • A low pass filtered signal
3. Select the maximum distance over which to compute the structure function based on conditions of the flow (e.g., expected max overturn) [r_max] in bin separation distances [n_max = r_max / r₀]
4. The structure function for a data segment can be calculated using either a bin-centred difference or a forward-difference scheme
5. 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
6. For a forward-difference scheme
   1. start with n being the lowest number bin of the range over which the structure function is to be evaluated (number of bins in range must exceed n_max
      1. start with [math]\displaystyle{ \delta }[/math] = 1
      2. 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 bin n and bin n + [math]\displaystyle{ \delta }[/math]:
         
         D(n, [math]\displaystyle{ \delta }[/math]) = [math]\displaystyle{ \langle }[/math] [v’(n, t) - v’(n+[math]\displaystyle{ \delta }[/math], 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. increment [math]\displaystyle{ \delta }[/math] and repeat steps until [math]\displaystyle{ \delta }[/math] = n_max or n + [math]\displaystyle{ \delta }[/math] exceeds the last bin of the range over which the structure function is to be evaluated
   2. increment n and repeat steps until n + 1 is the last bin of the range over which the structure function is to be evaluated
7. Including D(n, [math]\displaystyle{ \delta }[/math]) for [math]\displaystyle{ \delta }[/math] = 1 may be inappropriate since the velocity estimates from adjacent bins are not wholly independent, therefore the impact of its inclusion should be evaluated
8. The number of instances when the squared velocity difference is evaluated for each bin n and separation distance [math]\displaystyle{ \delta }[/math]r₀ and their distribution are potential quality control metrics
9. The impact of additional quality criteria can also be tested e.g. valid data requirements for all intermediate separation distances, so for a forward-difference scheme with n = 2 and [math]\displaystyle{ \delta }[/math] = 5, require all data in bins 2 to 7 to meet Level 1 QC requirements for the profile to be included when averaging to calculate D(n, [math]\displaystyle{ \delta }[/math])
10. Having calculated the segment D(n, [math]\displaystyle{ \delta }[/math]) for the appropriate range of bins and [math]\displaystyle{ \delta }[/math]r₀ separation distances, a regression of D(n, [math]\displaystyle{ \delta }[/math]) against ([math]\displaystyle{ \delta }[/math]r₀)^2/3 is then undertaken.
    1. If D(n, [math]\displaystyle{ \delta }[/math]) was evaluated using a forward-difference scheme, the regression is done for the combined data from all bins in the selected range, hence the maximum number of D(n, [math]\displaystyle{ \delta }[/math]) values for each separation distance will be the number of bins in the range less 1 for [math]\displaystyle{ \delta }[/math] = 1, reducing by 1 for each increment in [math]\displaystyle{ \delta }[/math], with the regression ultimately yielding a single [math]\displaystyle{ \varepsilon }[/math] value for the data segment
    2. If D(n, [math]\displaystyle{ \delta }[/math]) was evaluated using a bin-centred difference scheme, the regression can either be done:
       1. for each bin individually, with a single D(n, [math]\displaystyle{ \delta }[/math]) for each separation distance, ultimately yielding an [math]\displaystyle{ \varepsilon }[/math] for each bin; or
       2. by combining the data for all of the bins, with each separation distance having a D(n, [math]\displaystyle{ \delta }[/math]) value for each bin, with the regression again ultimately yielding a single [math]\displaystyle{ \varepsilon }[/math] value for the data segment
    3. The regression is typically done as a least-squares fit, either as:
       
       D = a₀ + a₁ ([math]\displaystyle{ \delta }[/math]r₀)^2/3; or as
       D = a₀ + a₁ ([math]\displaystyle{ \delta }[/math]r₀)^2/3 + a₃ (([math]\displaystyle{ \delta }[/math]r₀)^2/3)³
       
       the former being the canonical approach assuming no non-turbulent velocity difference between bins in v’, whilst the latter seeks to isolate the turbulent contribution from velocity differences between bins due to the orbital velocity forced by surface waves or any residual velocity shear retained due to the oscillatory motion of the ADCP on a mooring, both of which result in a velocity difference between bins which varies linearly with separation distance and hence contributes to D(n, [math]\displaystyle{ \delta }[/math]) as ([math]\displaystyle{ \delta }[/math]r₀)²
    4. The coefficient a₁ is then used to calculate [math]\displaystyle{ \varepsilon }[/math] as
       
       [math]\displaystyle{ \varepsilon }[/math] = (a₁ / C₂)^2/3
       
       where C₂ is an empirical constant, typically taken as 2.0 or 2.1

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