Linear optics measurement and correction - I 111
becomes rank deficient. If this happens to the Jacobian matrix of the leastsquare problem, there is a degeneracy involving the corresponding fitting parameters. Figure 4.5 (left) shows the SVs of the Jacobian matrix for the 78
fitting quadrupoles of SPEAR3. The two smallest SVs correspond to the two
pairs of correlated quadrupoles. The V -vectors for these two SV modes are
shown in the right plot. The last SV mode (V 78 ) primarily represents a neardegeneracy involving quadrupole parameters 60 (QDX) and 63 (QFX) - when
these two quadrupoles change in opposite directions, the net impact to the
orbit response matrix is small, hence the small SV. Similarly, the second-tolast SV mode (V 77 ) represents another degeneracy the involves quadrupole
parameters 59 (QDX) and 62 (QFX). Quadrupole parameters 76-78 are three
quadrupole magnets in the chicane straight, which are right between the two
pairs of QDX-QFX magnets. There are also correlations between these parameters.
The orbit response matrix data cannot effectively constrain the potential variations of the fitting parameters in the patterns represented by the
V -vectors that correspond to small SVs. As shown in Eq. (4.31), a small projection of the residual vector to such an SV mode will generate a large step
change to the fitting parameter along the V -vector. Conversely, a large step
in the V -vector of the mode only causes a small change to the χ
2 function.
SV modes with small SVs may be referred to as under-constrained directions.
Consequently, parameters that have large footprints in the under-constrained
directions have large error bars, as indicated in Eq. (4.35). In the SPEAR3
case, because of the correlations as discussed in the above, there are large error
bars to quadrupole parameters 59, 60, 62, 63, 76, and 78 (see Figure 4.3).
The error bar estimation by Eq. (4.35) only considers the random noise in
BPM measurements. Real orbit response matrix data may contain additional
errors due to orbit drifts and machine optics fluctuations and other systematic
errors. Before the fitting converges, the Jacobian matrix calculated with the
model differs from the Jacobian matrix for the measured optics. For example,
the differences between the SVs of the Jacobian matrices of the ideal model
and that of the actual lattice are up to 60% in the SPEAR3 test case. Furthermore, the expansion of the objective function based on the linearized model
of the residual vector (see Eq. (4.29)) may not be accurate when the present
solution is far from the minimum. Therefore, during the Gauss-Newton iterations, especially the initial iteration, there could be erroneous projections
to the SV modes with small SVs. These projections cause significant changes
to the fitting parameters, ∆p. The resulting solution for the iteration could
further deviate from the actual minimum, instead of converging toward it. In
some cases, the solution will eventually converge to the minimum. However,
if the deviation is large, the fitting method may fail to converge. In an extreme case, the changes to the quadrupole gradients can be so large such that
the lattice is not stable (e.g., no closed orbit can be found with the updated
model) and hence no further iteration can be carried out [49, 52].
The approach often used in the Gauss-Newton method to deal with small
SVs is to set a cut-off threshold and to eliminate all the SVs below the
becomes rank deficient. If this happens to the Jacobian matrix of the leastsquare problem, there is a degeneracy involving the corresponding fitting parameters. Figure 4.5 (left) shows the SVs of the Jacobian matrix for the 78
fitting quadrupoles of SPEAR3. The two smallest SVs correspond to the two
pairs of correlated quadrupoles. The V -vectors for these two SV modes are
shown in the right plot. The last SV mode (V 78 ) primarily represents a neardegeneracy involving quadrupole parameters 60 (QDX) and 63 (QFX) - when
these two quadrupoles change in opposite directions, the net impact to the
orbit response matrix is small, hence the small SV. Similarly, the second-tolast SV mode (V 77 ) represents another degeneracy the involves quadrupole
parameters 59 (QDX) and 62 (QFX). Quadrupole parameters 76-78 are three
quadrupole magnets in the chicane straight, which are right between the two
pairs of QDX-QFX magnets. There are also correlations between these parameters.
The orbit response matrix data cannot effectively constrain the potential variations of the fitting parameters in the patterns represented by the
V -vectors that correspond to small SVs. As shown in Eq. (4.31), a small projection of the residual vector to such an SV mode will generate a large step
change to the fitting parameter along the V -vector. Conversely, a large step
in the V -vector of the mode only causes a small change to the χ
2 function.
SV modes with small SVs may be referred to as under-constrained directions.
Consequently, parameters that have large footprints in the under-constrained
directions have large error bars, as indicated in Eq. (4.35). In the SPEAR3
case, because of the correlations as discussed in the above, there are large error
bars to quadrupole parameters 59, 60, 62, 63, 76, and 78 (see Figure 4.3).
The error bar estimation by Eq. (4.35) only considers the random noise in
BPM measurements. Real orbit response matrix data may contain additional
errors due to orbit drifts and machine optics fluctuations and other systematic
errors. Before the fitting converges, the Jacobian matrix calculated with the
model differs from the Jacobian matrix for the measured optics. For example,
the differences between the SVs of the Jacobian matrices of the ideal model
and that of the actual lattice are up to 60% in the SPEAR3 test case. Furthermore, the expansion of the objective function based on the linearized model
of the residual vector (see Eq. (4.29)) may not be accurate when the present
solution is far from the minimum. Therefore, during the Gauss-Newton iterations, especially the initial iteration, there could be erroneous projections
to the SV modes with small SVs. These projections cause significant changes
to the fitting parameters, ∆p. The resulting solution for the iteration could
further deviate from the actual minimum, instead of converging toward it. In
some cases, the solution will eventually converge to the minimum. However,
if the deviation is large, the fitting method may fail to converge. In an extreme case, the changes to the quadrupole gradients can be so large such that
the lattice is not stable (e.g., no closed orbit can be found with the updated
model) and hence no further iteration can be carried out [49, 52].
The approach often used in the Gauss-Newton method to deal with small
SVs is to set a cut-off threshold and to eliminate all the SVs below the