Bin-centred difference scheme: Difference between revisions
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# start at bin <math>n = \frac{n_{\text{rmax}}}{2} + 1</math> | # 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 <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[ | ## 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[b^\prime(n+\frac{\delta}{2},\ t) - b^\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 <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_{\text{lo}}(n, \delta, t) = | ## 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_{\text{lo}}(n, \delta, t) = b^\prime(n+\text{floor}\left(\frac{\delta}{2}\right),\ t) - b^\prime(n-\text{ceil}\left(\frac{\delta}{2}\right),\ t)</math> <br/> <math>\Delta_{\text{hi}}(n, \delta, t) = b^\prime(n+\text{ceil}\left(\frac{\delta}{2}\right),\ t) - b^\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_{\text{lo}}(n, \delta, t)^2 + \Delta_{\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 = 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> | # increment <math>n</math> and repeat steps until <math>n + \frac{n_{\text{rmax}}}{2}</math> exceeds the bin number for which valid <math>b^\prime</math> are available | ||
See [[Example bin-centred difference | example bin-centred difference calculation]] for more detail regarding the calculation | See [[Example bin-centred difference | example bin-centred difference calculation]] for more detail regarding the calculation | ||
Latest revision as of 12:56, 23 May 2022
For a bin-centred difference scheme:
- start at bin <math>n = \frac{n_{\text{rmax}}}{2} + 1</math>
- start with <math>\delta</math> = 1
- 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>:
<math>D(n, \delta) = \Big\langle \big[b^\prime(n+\frac{\delta}{2},\ t) - b^\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>\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>:
<math>\Delta_{\text{lo}}(n, \delta, t) = b^\prime(n+\text{floor}\left(\frac{\delta}{2}\right),\ t) - b^\prime(n-\text{ceil}\left(\frac{\delta}{2}\right),\ t)</math>
<math>\Delta_{\text{hi}}(n, \delta, t) = b^\prime(n+\text{ceil}\left(\frac{\delta}{2}\right),\ t) - b^\prime(n-\text{floor}\left(\frac{\delta}{2}\right),\ t)</math>
where <math>\text{ceil}</math> and <math>\text{floor}</math> indicate the upper and lower integer value respectively, then
<math>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>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 = 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>b^\prime</math> are available
See example bin-centred difference calculation for more detail regarding the calculation
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