Processing your ADCP data using structure function techniques

To calculate the dissipation rate at a specific range bin and a specific time ensemble:

1. Extract or compute the along-beam bin center separation [[math]\displaystyle{ r_0 }[/math]] based on the instrument geometry
2. Calculate the along-beam velocity fluctuation time-series in each bin, [[math]\displaystyle{ v’(n, t) }[/math]] from the along-beam velocity data that has met the QC criteria (i.e. the data in Level 2 of the netcdf file)
3. Select the maximum distance ([math]\displaystyle{ r_{max} }[/math]) over which to compute the structure function based on conditions of the flow (e.g., expected max overturn). The corresponding number of bins are [[math]\displaystyle{ n_{max} = r_{max} / \delta r_0 }[/math]], where [math]\displaystyle{ \delta r_{0} }[/math] is the radial separation between bins.
4. Calculate the structure function [math]\displaystyle{ D }[/math] for all possible bin separations [math]\displaystyle{ \delta }[/math] using either a bin-centred difference scheme or a forward-difference scheme. Some things to consider are: [SHOULD WE NEED TO INCLUDE THESE HERE?]
   • Including [math]\displaystyle{ D(n,\delta) }[/math] for [math]\displaystyle{ \delta=1 }[/math] may be inappropriate since the velocity estimates from adjacent bins are not wholly independent, therefore the impact of its inclusion should be evaluated
   • Keep a record of the number of instances when the squared velocity difference is evaluated for each bin [math]\displaystyle{ n }[/math] and separation distance [math]\displaystyle{ \delta r_{0} }[/math] and their distribution because they are potential quality control metrics
   • 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 [math]\displaystyle{ n=2 }[/math] and [math]\displaystyle{ \delta=5 }[/math], require all data in bins 2 to 7 to meet Level 1 QC requirements for the profile to be included when averaging to calculate [math]\displaystyle{ D(n,\delta) }[/math]
5. Perform a regression of [math]\displaystyle{ D(n,\delta) }[/math] against [math]\displaystyle{ (\delta r_0)^{2/3} }[/math] for the appropriate range of bins and [math]\displaystyle{ \delta }[/math]r₀ separation distances. [THE FOLLOWING ITEMS ARE CONFUSING. SINCE THIS IS BEST PRACTICE, CAN WE JUST RECOMMEND ONE METHOD?]
   1. If [math]\displaystyle{ D(n,\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 [math]\displaystyle{ D(n, \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 [math]\displaystyle{ D(n,\delta) }[/math] was evaluated using a bin-centred difference scheme, the regression can either be done:
      • for each bin individually, with a single [math]\displaystyle{ D(n, \delta) }[/math] for each separation distance, ultimately yielding an [math]\displaystyle{ \varepsilon }[/math] for each bin; or
      • by combining the data for all of the bins, with each separation distance having a [math]\displaystyle{ D(n, \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:
      
      [math]\displaystyle{ D = a_0 + a_1 (\delta r_0)^{2/3} }[/math]; or as
      [math]\displaystyle{ D = a_0 + a_1 (\delta r_0)^{2/3}+a_3((\delta r_0)^{2/3})^3 }[/math]
      
      the former being the canonical method that excludes non-turbulent velocity differences between bins, whereas the latter is a modified method that includes non-turbulent velocity differences between bins due to any oscillatory signal (e.g. surface waves, motion of the ADCP on a mooring).
6. Use the coefficient [math]\displaystyle{ a_1 }[/math] to calculate [math]\displaystyle{ \varepsilon }[/math] as
   
   [math]\displaystyle{ \varepsilon = \left(\frac{a_1}{C_2}\right)^{2/3} }[/math]
   
   where [math]\displaystyle{ C_2 }[/math] is an empirical constant, typically taken as 2.0 or 2.1 [LINK TO A CONCEPTS OR FUNDAMENTALS PAGE ABOUT THIS].
7. Use the coefficient [math]\displaystyle{ a_1 }[/math] (the intercept of the regression) to estimate the noise of the velocity observations and compare to the expected value based on the instrument settings. [MOVE TO QA2 STEPS?]

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