Compute the spectra - Revision history

CynthiaBluteau: /* Spectral averaging techniques */

2022-07-11T15:43:02Z

Spectral averaging techniques

← Older revision		Revision as of 15:43, 11 July 2022
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	The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]]. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality and whether they are located in the inertial subrange, respectively.		The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]]. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality and whether they are located in the inertial subrange, respectively.

	~~{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}~~

	[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. In red, the fft-length was halved to 16 s (43 degrees of freedom), while the third example (in purple) uses a combination of block and band averaging with the same number of blocks as the first example (black) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. Note: the lowest frequencies of this spectra example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating <math>\varepsilon</math>]]		[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. In red, the fft-length was halved to 16 s (43 degrees of freedom), while the third example (in purple) uses a combination of block and band averaging with the same number of blocks as the first example (black) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. Note: the lowest frequencies of this spectra example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating <math>\varepsilon</math>]]

CynthiaBluteau: Added wikilcnks

2022-07-11T15:25:49Z

Added wikilcnks

← Older revision		Revision as of 15:25, 11 July 2022
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	}}</ref>		}}</ref>
	# Convert the spectrum from the time domain to the space domain using the mean speed past the sensor {{FontColor\|fg=white\|bg=red\|text= only for steady flows, not required for surface wave analysis}}		# Convert the spectrum from the time domain to the space domain using the mean speed past the sensor {{FontColor\|fg=white\|bg=red\|text= only for steady flows, not required for surface wave analysis}}
	# Compute degrees of freedom (dof) and confidence intervals of the final spectra <ref name="EmeryThomson2001"/> based on the assumption that the spectra observations are <math>\chi~~-squared~~</math> distributed i.e., the turbulent velocities are gaussian (normally distributed).		# Compute degrees of freedom (dof) and confidence intervals of the final spectra <ref name="EmeryThomson2001"/> based on the assumption that the spectra observations are <math>\chi</math>-squared distributed i.e., the turbulent velocities are gaussian (normally distributed).

	==<span id="specavg">Spectral averaging</span> techniques==		==<span id="specavg">Spectral averaging</span> techniques==

CynthiaBluteau: /* Spectral averaging techniques */

2022-07-11T15:25:08Z

Spectral averaging techniques

← Older revision		Revision as of 15:25, 11 July 2022
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	==<span id="specavg">Spectral averaging</span> techniques==		==<span id="specavg">Spectral averaging</span> techniques==
	Each segment is often subdivided into smaller [[#fftlength\|fft-length]] long chunks (50% overlap), which are then windowed before estimating numerous spectra (FFT) that are block-averaged for increased statistical significance. Another averaging strategy is band-averaging spectra in the frequency domain, which allows the [[Segmenting datasets\|segment length]] to be the same as the [[#fftlength\|fft-length]]. A combination of both strategies is also possible. The final strategy depends on whether you need increased statistical significance for correcting motion-contaminated spectra using ~~{{FontColor\|fg=white\|bg=red~~\|~~text=~~ cospectral methods}}, and the lowest frequencies (wavenumbers) you want to resolve.		Each segment is often subdivided into smaller [[#fftlength\|fft-length]] long chunks (50% overlap), which are then windowed before estimating numerous spectra (FFT) that are block-averaged for increased statistical significance. Another averaging strategy is band-averaging spectra in the frequency domain, which allows the [[Segmenting datasets\|segment length]] to be the same as the [[#fftlength\|fft-length]]. A combination of both strategies is also possible. The final strategy depends on whether you need increased statistical significance for correcting motion-contaminated spectra using [[Velocity decontamination by cospectral methods\|cospectral methods]], and the lowest frequencies (wavenumbers) you want to resolve.

	The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]]. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality and whether they are located in the inertial subrange, respectively.		The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]]. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality and whether they are located in the inertial subrange, respectively.

CynthiaBluteau: /* Spectral averaging techniques */

2022-07-11T14:37:27Z

Spectral averaging techniques

← Older revision		Revision as of 14:37, 11 July 2022
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	==<span id="specavg">Spectral averaging</span> techniques==		==<span id="specavg">Spectral averaging</span> techniques==
	~~The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]] (unless the spectra are band-avg).~~ Each segment is often subdivided into smaller [[#fftlength\|fft-length]] long chunks (50% overlap), which are then windowed before estimating numerous spectra (FFT) that are block-averaged for increased statistical significance. Another averaging strategy is band-averaging spectra in the frequency domain, which allows the [[Segmenting datasets\|segment length]] to be the same as the [[#fftlength\|fft-length]]. A combination of both strategies is also possible. The final strategy depends on whether you need increased statistical significance for correcting motion-contaminated spectra using {{FontColor\|fg=white\|bg=red\|text= cospectral methods}}, and the lowest frequencies (wavenumbers) you want to resolve. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these ~~large~~ frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality.		Each segment is often subdivided into smaller [[#fftlength\|fft-length]] long chunks (50% overlap), which are then windowed before estimating numerous spectra (FFT) that are block-averaged for increased statistical significance. Another averaging strategy is band-averaging spectra in the frequency domain, which allows the [[Segmenting datasets\|segment length]] to be the same as the [[#fftlength\|fft-length]]. A combination of both strategies is also possible. The final strategy depends on whether you need increased statistical significance for correcting motion-contaminated spectra using {{FontColor\|fg=white\|bg=red\|text= cospectral methods}}, and the lowest frequencies (wavenumbers) you want to resolve.

			The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]]. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality and whether they are located in the inertial subrange, respectively.

	{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}		{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}

CynthiaBluteau at 14:35, 11 July 2022

2022-07-11T14:35:53Z

← Older revision		Revision as of 14:35, 11 July 2022
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	# Convert the spectrum from the time domain to the space domain using the mean speed past the sensor {{FontColor\|fg=white\|bg=red\|text= only for steady flows, not required for surface wave analysis}}		# Convert the spectrum from the time domain to the space domain using the mean speed past the sensor {{FontColor\|fg=white\|bg=red\|text= only for steady flows, not required for surface wave analysis}}
	# Compute degrees of freedom (dof) and confidence intervals of the final spectra <ref name="EmeryThomson2001"/> based on the assumption that the spectra observations are <math>\chi-squared</math> distributed i.e., the turbulent velocities are gaussian (normally distributed).		# Compute degrees of freedom (dof) and confidence intervals of the final spectra <ref name="EmeryThomson2001"/> based on the assumption that the spectra observations are <math>\chi-squared</math> distributed i.e., the turbulent velocities are gaussian (normally distributed).


	==<span id="specavg">Spectral averaging</span> techniques==		==<span id="specavg">Spectral averaging</span> techniques==

CynthiaBluteau: Fixed formatting

2022-07-11T14:35:34Z

Fixed formatting

← Older revision		Revision as of 14:35, 11 July 2022
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	To compute the spectrum of the turbulent velocity fluctuations, you need to:		To compute the spectrum of the turbulent velocity fluctuations, you need to:

	# Determine appropriate [[#fftlength\|fft-length]] and [[#specavg\|spectral averaging]] for each data [[Segmenting datasets\|segment]]		# Determine appropriate [[#fftlength\|fft-length]] and [[#specavg\|spectral averaging]] for each data [[Segmenting datasets\|segment]]
	# Compute the spectrum using standard techniques <ref name="EmeryThomson2001">{{Cite journal		# Compute the spectrum using standard techniques <ref name="EmeryThomson2001">{{Cite journal

CynthiaBluteau: /* Spectral averaging techniques */

2022-07-11T14:20:52Z

Spectral averaging techniques

← Older revision		Revision as of 14:20, 11 July 2022
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	{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}		{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}

	[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. ~~The~~ fft-length was halved to 16 s ~~in red~~ (43 degrees of freedom), while the third example (purple) uses a combination of block and band averaging~~. The blocks were~~ the same as the first example (~~32-s long~~) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. ~~The~~ lowest frequencies in this example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating <math>\varepsilon</math>]]		[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. In red, the fft-length was halved to 16 s (43 degrees of freedom), while the third example (in purple) uses a combination of block and band averaging with the same number of blocks as the first example (black) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. Note: the lowest frequencies of this spectra example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating <math>\varepsilon</math>]]

	==References==		==References==

CynthiaBluteau: /* Spectral averaging techniques */

2022-07-11T14:18:26Z

Spectral averaging techniques

← Older revision		Revision as of 14:18, 11 July 2022
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	{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}		{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}

	[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. The fft-length was halved to 16 s in red (43 degrees of freedom), while the third example (purple) uses a combination of block and band averaging. The blocks were the same as the first example (32-s long) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58.]]		[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. The fft-length was halved to 16 s in red (43 degrees of freedom), while the third example (purple) uses a combination of block and band averaging. The blocks were the same as the first example (32-s long) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. The lowest frequencies in this example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating <math>\varepsilon</math>]]

	==References==		==References==

CynthiaBluteau at 14:16, 11 July 2022

2022-07-11T14:16:11Z

← Older revision		Revision as of 14:16, 11 July 2022
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	{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}		{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}

	[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. The fft-length was halved to 16 s in red (43 degrees of freedom), while the third example (purple) uses a combination of block and band averaging. The blocks were the same as the first example (32-s long) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. ~~The degrees of freedom and statistical confidence interval were estimated using the methods described in <ref name="EmeryThomson2001"/> and <ref name="Priestly1981"/>~~]]		[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. The fft-length was halved to 16 s in red (43 degrees of freedom), while the third example (purple) uses a combination of block and band averaging. The blocks were the same as the first example (32-s long) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58.]]

	==References==		==References==

CynthiaBluteau at 14:14, 11 July 2022

2022-07-11T14:14:51Z

← Older revision		Revision as of 14:14, 11 July 2022
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	\|year= 1981		\|year= 1981
	\|doi=(ISBN)0125649010		\|doi=(ISBN)0125649010
	}}</ref>~~based on the assumption that the spectra observations are <math>\chi-squared</math> distributed i.e., the turbulent velocities are gaussian (normally distributed).~~		}}</ref>
	# Convert the spectrum from the time domain to the space domain using the mean speed past the sensor {{FontColor\|fg=white\|bg=red\|text= only for steady flows, not required for surface wave analysis}}		# Convert the spectrum from the time domain to the space domain using the mean speed past the sensor {{FontColor\|fg=white\|bg=red\|text= only for steady flows, not required for surface wave analysis}}
	# Compute degrees of freedom (dof) and confidence intervals of the final spectra <ref name="EmeryThomson2001"/>		# Compute degrees of freedom (dof) and confidence intervals of the final spectra <ref name="EmeryThomson2001"/> based on the assumption that the spectra observations are <math>\chi-squared</math> distributed i.e., the turbulent velocities are gaussian (normally distributed).


	==<span id="specavg">Spectral averaging</span> techniques==		==<span id="specavg">Spectral averaging</span> techniques==
	The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]] (unless the spectra are band-avg). Each segment is often subdivided into smaller [[#fftlength\|fft-length]] long chunks (50% overlap), which are then windowed before estimating numerous spectra (FFT) that are block-averaged for increased statistical significance. Another averaging strategy is band-averaging spectra in the frequency domain, which allows the [[Segmenting datasets\|segment length]] to be the same as the [[#fftlength\|fft-length]]. A combination of both strategies is also possible. The final strategy depends on whether you need increased statistical significance for correcting motion-contaminated spectra using ~~methods based on~~ {{FontColor\|fg=white\|bg=red\|text= ~~squared coherency~~}}, and the lowest frequencies (wavenumbers) you want to resolve. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these large frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality.		The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]] (unless the spectra are band-avg). Each segment is often subdivided into smaller [[#fftlength\|fft-length]] long chunks (50% overlap), which are then windowed before estimating numerous spectra (FFT) that are block-averaged for increased statistical significance. Another averaging strategy is band-averaging spectra in the frequency domain, which allows the [[Segmenting datasets\|segment length]] to be the same as the [[#fftlength\|fft-length]]. A combination of both strategies is also possible. The final strategy depends on whether you need increased statistical significance for correcting motion-contaminated spectra using {{FontColor\|fg=white\|bg=red\|text= cospectral methods}}, and the lowest frequencies (wavenumbers) you want to resolve. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these large frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality.

	{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}		{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}

← Older revision		Revision as of 15:43, 11 July 2022
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	The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]]. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality and whether they are located in the inertial subrange, respectively.		The spectrum's lowest resolved frequency and final resolution are the inverses of the [[#fftlength\| fft-length]]. The [[#fftlength\|fft-length]] dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate <math>\varepsilon</math> depends on the measurement quality and whether they are located in the inertial subrange, respectively.

	~~{{FontColor\|fg=white\|bg=red\|text= Remove redundant info from [[Segmenting datasets]], and add references to figure summary page}}~~

	[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. In red, the fft-length was halved to 16 s (43 degrees of freedom), while the third example (in purple) uses a combination of block and band averaging with the same number of blocks as the first example (black) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. Note: the lowest frequencies of this spectra example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating <math>\varepsilon</math>]]		[[File:Spectra computation.png\|thumbnail\|800px\|Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. In red, the fft-length was halved to 16 s (43 degrees of freedom), while the third example (in purple) uses a combination of block and band averaging with the same number of blocks as the first example (black) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. Note: the lowest frequencies of this spectra example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating <math>\varepsilon</math>]]