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>r_0</math>] based on the instrument geometry
2. Calculate the along-beam velocity fluctuation time-series in each bin, [<math>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>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 is [<math>n_{max} = r_{max} / r_0</math>]
4. Calculate the structure function <math>D</math> for all possible bin separations <math>\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>D(n,\delta)</math> for <math>\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>n</math> and separation distance <math>\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>n=2</math> and <math>\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>D(n,\delta)</math>
5. Perform a regression of <math>D(n,\delta)</math> against <math>(\delta r_0)^{2/3}</math> for the appropriate range of bins and <math>\delta</math>r₀ separation distances. [THE FOLLOWING ITEMS ARE CONFUSING. SINCE THIS IS BEST PRACTICE, CAN WE JUST RECOMMEND ONE METHOD?]
   1. If <math>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>D(n, \delta)</math> values for each separation distance will be the number of bins in the range less 1 for <math>\delta</math> = 1, reducing by 1 for each increment in <math>\delta</math>, with the regression ultimately yielding a single <math>\varepsilon</math> value for the data segment
   2. If <math>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>D(n, \delta)</math> for each separation distance, ultimately yielding an <math>\varepsilon</math> for each bin; or
      • by combining the data for all of the bins, with each separation distance having a <math>D(n, \delta)</math> value for each bin, with the regression again ultimately yielding a single <math>\varepsilon</math> value for the data segment
   3. The regression is typically done as a least-squares fit, either as:
      
      <math>D = a_0 + a_1 (\delta r_0)^{2/3}</math>; or as
      <math>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>a_1</math> to calculate <math>\varepsilon</math> as
   
   <math>\varepsilon = \left(\frac{a_1}{C_2}\right)^{2/3}</math>
   
   where <math>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>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?]

PERHAPS WE CAN INCLUDE A FIGURE LIKE THIS TO HELP DEFINE VARIABLES.

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