Processing your ADCP data using structure function techniques: Difference between revisions

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


# Extract or compute the along-beam bin center separation [r0] based on the instrument geometry
[[File:ADCPschematic SF.png|thumb|Schematic showing along-beam distance <math> r </math> and radial velocities. ]]
# 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
## 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
## If using continuous sampling, calculations are dome over segments with a duration over which the turbulent flow statistics can assumed to be stationary
## 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''
# Select the maximum distance over which to compute the structure function based on conditions of the flow (e.g., expected max overturn) [r<sub>max</sub>] in bin separation distances [n<sub>max</sub> = r<sub>max</sub> / r<sub>0</sub>]
# The structure function for a data segment can be calculated using either a '''bin-centred difference''' or a '''forward-difference''' scheme
# For a '''bin-centred difference''' scheme
## start at bin n = (n<sub>max</sub> / 2) + 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 '''''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
### increment <math>\delta</math> and repeat steps until <math>\delta</math> = n<sub>max</sub>
## increment n and repeat steps until n + (n<sub>max</sub> / 2) exceeds the bin number for which valid v’ are available
# For a '''forward-difference''' scheme
## 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<sub>max</sub>
### start with <math>\delta</math> = 1
### compute the second order structure function D(n,<math>\delta</math>) as the segment mean of the square of the velocity difference between the bin n and bin n + <math>\delta</math>: <br/><br /> D(n, <math>\delta</math>) = <math>\langle</math> [v’(n, t) - v’(n+<math>\delta</math>, 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
### increment <math>\delta</math> and repeat steps until <math>\delta</math> = n<sub>max</sub> or n + <math>\delta</math> exceeds the last bin of the range over which the structure function is to be evaluated
## increment n and repeat steps until n + 1 is the last bin of the range over which the structure function is to be evaluated
# Including D(n, <math>\delta</math>) for <math>\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
# The number of instances when the squared velocity difference is evaluated for each bin n and separation distance <math>\delta</math>r<sub>0</sub> and their distribution are potential quality control metrics


'''[IN PROGRESS]'''
# Extract or compute the [[along-beam bin center separation]] [<math>\delta r_0</math>] based on the instrument geometry
## Compute the second order structure function D(z,r) = mean-square of the velocity fluctuation difference:  D(z,2*r0) = mean(v’(z+r<sub>0</sub>)-v’(z-r<sub>0</sub>))<sup>2</sup>  
# Calculate the [[along-beam velocity fluctuation]] time-series in each bin <math>n</math>, where [<math>b’(n, t_s)</math>]  from the along-beam velocity data that has met the QC criteria (i.e. the data in Level 2 of the netcdf file).  Note <math> t_s </math> is the timeseries index within a segment.
## Repeat steps 5-6 for all pairs of bins where the separation distance between bins r <= r<sub>max</sub>  
# 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, spectral range corresponding to <math> k^{-5/3} </math>). The corresponding number of bins is [<math>n_{\text{rmax}} = r_{max} / \delta r_0</math>]
# Check if all points involved in the differencing to contain good data, e.g. If I were starting from bin=2, with a maximum separation distance of 5, I required all data in bins 2 to 7 to meet QC requirements (usually just use correlation threshold). If yes, continue with the next step. If not, exclude this profile.
# Calculate the structure function <math>D_{ll}</math> for all possible bin separations <math>\delta</math> within <math>r_{max}</math> using either a [[bin-centred difference scheme]] or a [[forward-difference]] scheme.
# With valid, contiguous data points, fit a line to the form D(z,r) = N + Ar<sup>2</sup>/3 to estimate values for A and N where A = Cv<sup>2</sup>ε<sup>2</sup>/3  and N is an estimate of the uncertainty due to noise.  
# Perform a regression of <math>D_{ll}(n,\delta)</math> against <math>(\delta r)^{2/3}</math> for the appropriate range of bins and <math>\delta</math>r</sub> separation distances. Be aware of [[Regressing structure function against bin separation | special considerations for forward-difference, center-difference schemes]] in setting up the regression calculation.   The regression is typically done as a least-squares fit, either as: <br /><br /> <math>D_{ll} = a_0 + a_1 (\delta r)^{2/3}</math>;
# Solve for ε using Cv<sup>2</sup> = 2.1
:: or as
# Repeat the steps in (5) – (9) for each bin until z<sub>b</sub> + r<sub>max</sub>/2 >= end of profile
:: <math>D_{ll} = a_0 + a_1 (\delta r)^{2/3}+a_3((\delta r)^{2/3})^3 </math> <br /><br /> the former being the [[canonical structure function method | canonical method]] that excludes non-turbulent velocity differences between bins, whereas the latter is a [[modified structure function method | 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).
<ol type="1" start=6>
<li> Use the coefficient <math>a_1</math> to calculate <math>\varepsilon</math> as <br /><br /> <math>\varepsilon = \left(\frac{a_1}{C_2}\right)^{2/3}</math> <br /><br /> where <math>C_2</math> is an [[ Structure function empirical constant | empirical constant]], typically taken as 2.0 or 2.1.
</ol>


----
Next step:  [[Final data review (QA2) | Apply quality-control on dissipation rates (QA2)]] <br></br>
Previous step:[[Raw data review (QA1) | Apply quality-control on velocity time series data (QA1)]]<br></br>
Return to [[ADCP structure function flow chart| ADCP Flow Chart front page]]
Return to [[ADCP structure function flow chart| ADCP Flow Chart front page]]
[[Category:Velocity profilers]]

Latest revision as of 15:51, 30 May 2022

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

Schematic showing along-beam distance [math]\displaystyle{ r }[/math] and radial velocities.
  1. Extract or compute the along-beam bin center separation [[math]\displaystyle{ \delta r_0 }[/math]] based on the instrument geometry
  2. Calculate the along-beam velocity fluctuation time-series in each bin [math]\displaystyle{ n }[/math], where [[math]\displaystyle{ b’(n, t_s) }[/math]] from the along-beam velocity data that has met the QC criteria (i.e. the data in Level 2 of the netcdf file). Note [math]\displaystyle{ t_s }[/math] is the timeseries index within a segment.
  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, spectral range corresponding to [math]\displaystyle{ k^{-5/3} }[/math]). The corresponding number of bins is [[math]\displaystyle{ n_{\text{rmax}} = r_{max} / \delta r_0 }[/math]]
  4. Calculate the structure function [math]\displaystyle{ D_{ll} }[/math] for all possible bin separations [math]\displaystyle{ \delta }[/math] within [math]\displaystyle{ r_{max} }[/math] using either a bin-centred difference scheme or a forward-difference scheme.
  5. Perform a regression of [math]\displaystyle{ D_{ll}(n,\delta) }[/math] against [math]\displaystyle{ (\delta r)^{2/3} }[/math] for the appropriate range of bins and [math]\displaystyle{ \delta }[/math]r separation distances. Be aware of special considerations for forward-difference, center-difference schemes in setting up the regression calculation. The regression is typically done as a least-squares fit, either as:

    [math]\displaystyle{ D_{ll} = a_0 + a_1 (\delta r)^{2/3} }[/math];
or as
[math]\displaystyle{ D_{ll} = a_0 + a_1 (\delta r)^{2/3}+a_3((\delta r)^{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).
  1. 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.


Next step: Apply quality-control on dissipation rates (QA2)

Previous step: Apply quality-control on velocity time series data (QA1)

Return to ADCP Flow Chart front page

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