Linear optics measurement and correction - II 129
Three BPMs are used in the above scheme as this is the minimum number
of BPMs in order to solve for α 1 and β 1 with Eq. (5.16). To obtain the beta
function at one BPM, typically its two immediate neighbors are used if the
phase advances satisfy the requirements. Sometimes an immediate neighbor
has to be skipped if the phase advance is too close to 0,
π
2 , or π.
Using multiple 3-BPM combinations with the two other BPMs going beyond the immediate neighbors can improve the beta measurement accuracy
as it brings in a statistical advantage [73]. The measured beta function will be
a weighted sum of the selected combinations. The weights are not equal because the phase advance measurement errors of the different combinations are
correlated through the common BPMs and also the systematic errors (due
to differences in the measured and model transfer matrices) depend on the
lattice section between the BPMs. The ideal weights should be derived from
the covariance matrix of the measured beta of all 3-BPM combinations, with
contributions from both the random phase noise and the systematic errors
included. In this case, the method is called the N-BPM method.
The beta functions measured with the 3-BPM or N-BPM methods are
not affected by BPM calibrations. By comparing the
√
β measured this way
to the betatron oscillation amplitudes, the BPM gains could be calibrated.
However, because the beta functions are derived from the phase measurement
and thus do not bring in any new information, including these beta functions
as input data (in addition to the phase advance measurement) in the linear
optics model fitting does not necessarily improve the fitting result.
5.2 GLOBAL ANALYSIS OF TURN-BY-TURN BPM DATA
In the harmonic analysis of the turn-by-turn BPM data, data on different
BPMs are analyzed separately. Because of the random noise and the effect
of the finite number of turns, the betatron frequency and phase advances
obtained from the BPMs may be inconsistent. The results can be substantially
improved if all BPM data are analyzed together. This will lead to significantly
improved accuracy not only by producing consistent results among all BPMs,
but also by taking advantage of the statistics offered by the multiple samples
of the same physical processes.
Model independent analysis (MIA) [65, 119] is such a method of global
analysis of all BPM data. The statistical analysis method it employs is the
principal component analysis (PCA) [97]. In the ideal scenario, through PCA,
the various oscillation harmonics in the raw BPM data can be automatically
identified and separated. All BPMs share the same time evolution of the harmonics and are thus inherently consistent. Typically there are only a few PCA
components that contain information of the beam motion. Since the random
noise is distributed over all PCA components, the signal-to-noise ratios in the
actual beam motion signals can be improved after the relevant PCA components are isolated. Because the term “model independent analysis” has a very
Three BPMs are used in the above scheme as this is the minimum number
of BPMs in order to solve for α 1 and β 1 with Eq. (5.16). To obtain the beta
function at one BPM, typically its two immediate neighbors are used if the
phase advances satisfy the requirements. Sometimes an immediate neighbor
has to be skipped if the phase advance is too close to 0,
π
2 , or π.
Using multiple 3-BPM combinations with the two other BPMs going beyond the immediate neighbors can improve the beta measurement accuracy
as it brings in a statistical advantage [73]. The measured beta function will be
a weighted sum of the selected combinations. The weights are not equal because the phase advance measurement errors of the different combinations are
correlated through the common BPMs and also the systematic errors (due
to differences in the measured and model transfer matrices) depend on the
lattice section between the BPMs. The ideal weights should be derived from
the covariance matrix of the measured beta of all 3-BPM combinations, with
contributions from both the random phase noise and the systematic errors
included. In this case, the method is called the N-BPM method.
The beta functions measured with the 3-BPM or N-BPM methods are
not affected by BPM calibrations. By comparing the
√
β measured this way
to the betatron oscillation amplitudes, the BPM gains could be calibrated.
However, because the beta functions are derived from the phase measurement
and thus do not bring in any new information, including these beta functions
as input data (in addition to the phase advance measurement) in the linear
optics model fitting does not necessarily improve the fitting result.
5.2 GLOBAL ANALYSIS OF TURN-BY-TURN BPM DATA
In the harmonic analysis of the turn-by-turn BPM data, data on different
BPMs are analyzed separately. Because of the random noise and the effect
of the finite number of turns, the betatron frequency and phase advances
obtained from the BPMs may be inconsistent. The results can be substantially
improved if all BPM data are analyzed together. This will lead to significantly
improved accuracy not only by producing consistent results among all BPMs,
but also by taking advantage of the statistics offered by the multiple samples
of the same physical processes.
Model independent analysis (MIA) [65, 119] is such a method of global
analysis of all BPM data. The statistical analysis method it employs is the
principal component analysis (PCA) [97]. In the ideal scenario, through PCA,
the various oscillation harmonics in the raw BPM data can be automatically
identified and separated. All BPMs share the same time evolution of the harmonics and are thus inherently consistent. Typically there are only a few PCA
components that contain information of the beam motion. Since the random
noise is distributed over all PCA components, the signal-to-noise ratios in the
actual beam motion signals can be improved after the relevant PCA components are isolated. Because the term “model independent analysis” has a very