Bin-centred difference scheme: Difference between revisions
From Atomix
No edit summary |
No edit summary |
||
| Line 6: | Line 6: | ||
## increment <math>\delta</math> and repeat steps until <math>\delta = n_{\text{rmax}}</math> | ## increment <math>\delta</math> and repeat steps until <math>\delta = n_{\text{rmax}}</math> | ||
# increment <math>n</math> and repeat steps until <math>n + \frac{n_{\text{rmax}}}{2}</math> exceeds the bin number for which valid <math>v^\prime</math> are available | # increment <math>n</math> and repeat steps until <math>n + \frac{n_{\text{rmax}}}{2}</math> exceeds the bin number for which valid <math>v^\prime</math> are available | ||
See [[Example bin-centred difference | example bin-centred difference calculation]] for more detail regarding the calculation | |||
Return to [[Processing your ADCP data using structure function techniques | Compute structure functions and dissipation estimates]] | Return to [[Processing your ADCP data using structure function techniques | Compute structure functions and dissipation estimates]] | ||
Revision as of 10:03, 15 November 2021
For a bin-centred difference scheme:
- start at bin [math]\displaystyle{ n = \frac{n_{\text{rmax}}}{2} + 1 }[/math]
- start with [math]\displaystyle{ \delta }[/math] = 1
- if [math]\displaystyle{ \delta }[/math] is even compute the second order structure function [math]\displaystyle{ D(n,\delta) }[/math] as the segment mean of the square of the velocity difference between the bins separated by distance [math]\displaystyle{ \delta r_0 }[/math] centered around bin [math]\displaystyle{ n }[/math]:
[math]\displaystyle{ D(n, \delta) = \Big\langle \big[v^\prime(n+\frac{\delta}{2},\ t) - v^\prime(n-\frac{\delta}{2},\ t)\big]^2 \Big\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 - if [math]\displaystyle{ \delta }[/math] is odd compute the second order structure function [math]\displaystyle{ D(n,\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 r_0 }[/math] centered on the upper and lower extent of bin [math]\displaystyle{ n }[/math]:
[math]\displaystyle{ \Delta_{\text{lo}}(n, \delta, t) = v^\prime(n+\text{floor}\left(\frac{\delta}{2}\right),\ t) - v^\prime(n-\text{ceil}\left(\frac{\delta}{2}\right),\ t) }[/math]
[math]\displaystyle{ \Delta_{\text{hi}}(n, \delta, t) = v^\prime(n+\text{ceil}\left(\frac{\delta}{2}\right),\ t) - v^\prime(n-\text{floor}\left(\frac{\delta}{2}\right),\ t) }[/math]
where [math]\displaystyle{ \text{ceil} }[/math] and [math]\displaystyle{ \text{floor} }[/math] indicate the upper and lower integer value respectively, then
[math]\displaystyle{ D(n, \delta) = \Bigg\langle \frac{\Delta_{\text{lo}}(n, \delta, t)^2 + \Delta_{\text{hi}}(n, \delta, t)^2}{2} \Bigg\rangle }[/math]
the angled brackets again indicating the mean across all [math]\displaystyle{ t }[/math] in the data segment yielding a velocity difference after the application of the Level 1 QC criteria - increment [math]\displaystyle{ \delta }[/math] and repeat steps until [math]\displaystyle{ \delta = n_{\text{rmax}} }[/math]
- increment [math]\displaystyle{ n }[/math] and repeat steps until [math]\displaystyle{ n + \frac{n_{\text{rmax}}}{2} }[/math] exceeds the bin number for which valid [math]\displaystyle{ v^\prime }[/math] are available
See example bin-centred difference calculation for more detail regarding the calculation
Return to Compute structure functions and dissipation estimates
