Bin-centred difference scheme: Difference between revisions
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Jmmcmillan (talk | contribs) Created page with "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''''' compu..." |
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For a '''bin-centred difference''' scheme: | For a '''bin-centred difference''' scheme: | ||
# start at bin n = | # start at bin <math>n = \frac{n_{\text{rmax}}}{2} + 1</math> | ||
## start with <math>\delta</math> = 1 | ## start with <math>\delta</math> = 1 | ||
## if <math>\delta</math> is '''''even''''' compute the second order structure function D(n, | ## if <math>\delta</math> is '''''even''''' compute the second order structure function <math>D(n,\delta)</math> as the segment mean of the square of the velocity difference between the bins separated by distance <math>\delta r_0</math> centered around bin <math>n</math>: <br /><br /><math>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> <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, | ## if <math>\delta</math> is '''''odd''''' compute the second order structure function <math>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>\delta r_0</math> centered on the upper and lower extent of bin <math>n</math>: <br/><br /> <math>\Delta v^\prime_{\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> <br/> <math>\Delta v^\prime_{\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> <br/><br /> where <math>\text{ceil}</math> and <math>\text{floor}</math> indicate the upper and lower integer value respectively, then <br/><br /> <math>D(n, \delta) = \Bigg\langle \frac{\Delta v^\prime_{\text{lo}}(n, \delta, t)^2 + \Delta v^\prime_{\text{hi}}(n, \delta, t)^2}{2} \Bigg\rangle</math> <br/><br /> the angled brackets again indicating the mean across all <math>t</math> 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> | ## 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 | |||
Revision as of 09:57, 11 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 v^\prime_{\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 v^\prime_{\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 v^\prime_{\text{lo}}(n, \delta, t)^2 + \Delta v^\prime_{\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
