Regressing structure function against bin separation

How the regressions are set up depends on the choice of differencing scheme, these are explained below. Also, the most common choice of fitting method is recommended as best practice, but alternatives do exist.

Forward-difference scheme regression

1. If [math]\displaystyle{ D_{ll}(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_{ll}(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

Bin-centered difference scheme regression

1. If [math]\displaystyle{ D_{ll}(n,\delta) }[/math] was evaluated using a bin-centered 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_{ll}(n, \delta) }[/math] value for each bin, with the regression again ultimately yielding a single [math]\displaystyle{ \varepsilon }[/math] value for the data segment.

Least-squares fit

1. The regression is typically done as a least-squares fit, either as:
   
   [math]\displaystyle{ D_{ll} = a_0 + a_1 (\delta r_0)^{2/3} }[/math]; or as
   [math]\displaystyle{ D_{ll} = 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).

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