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R. Frühwirth et al.
a homogeneous magnetic field, the path length can be determined analytically only
for propagation to cylinders with symmetry axis parallel to the field direction or to
planes orthogonal to the field direction. Otherwise, a Newton iteration or a parabolic
approximation has to be used to find the path length.
For track reconstruction purposes, the covariance matrix of the estimated track
parameters needs to be propagated along with the track parameters themselves. The
track propagator is often a non-linear function of the track parameters at the initial
surface, but the covariance matrix has to be transported under the assumption of
a linear track model. This procedure, called linear error propagation, is based on
a Taylor expansion of the track propagator, keeping only first-order terms. These
first-order terms, defining the Jacobians of the track model, are given by
F k|i =
∂q k
∂q i
˘
q i
,
where ˘
q i is the expansion point in surface i. For analytical track models, the
Jacobian is also analytical. If, for example, the magnetic field is homogeneous,
the general case of propagation to a plane of arbitrary spatial orientation uses a
curvilinear coordinate frame moving along with the trajectory as a means of deriving
the required Jacobians [39].
In the general case of a non-analytical track model, the Jacobians cannot be
computed analytically either. The most straightforward approach is to calculate the
relevant derivatives in a purely numerical way. The basis for these calculations
is a reference trajectory corresponding to the expansion point. In addition, five
other trajectories are created, corresponding to small variations in each of the
track parameters. By propagating these five trajectories to the destination surface,
numerical derivatives can be obtained. A potential disadvantage of such an approach
is its computational complexity, as six trajectories have to be propagated instead of a
single one. Much less computational load is introduced by transporting the Jacobian
terms in parallel to the track parameters during the Runge–Kutta integration [40, 41],
avoiding the need for propagating auxiliary trajectories.
The measurement model describes the functional dependence of the measured
quantities on the state vector at a detector surface k:
m k = h k (q k ).
The vector of measurements m k usually contains the measured coordinates, but may
contain also other quantities, e.g. measurements of direction or even momentum. In
a pixel detector or in a double-sided silicon strip detector, m k is two-dimensional; in
a one-sided strip detector, it is one-dimensional. In a drift chamber or a multi-wire
proportional chamber with several layers, the measurement may be a track segment
resulting from an internal track reconstruction. In this case the vector m k may be
four- or five-dimensional, depending on whether the curvature can be estimated or
not.
R. Frühwirth et al.
a homogeneous magnetic field, the path length can be determined analytically only
for propagation to cylinders with symmetry axis parallel to the field direction or to
planes orthogonal to the field direction. Otherwise, a Newton iteration or a parabolic
approximation has to be used to find the path length.
For track reconstruction purposes, the covariance matrix of the estimated track
parameters needs to be propagated along with the track parameters themselves. The
track propagator is often a non-linear function of the track parameters at the initial
surface, but the covariance matrix has to be transported under the assumption of
a linear track model. This procedure, called linear error propagation, is based on
a Taylor expansion of the track propagator, keeping only first-order terms. These
first-order terms, defining the Jacobians of the track model, are given by
F k|i =
∂q k
∂q i
˘
q i
,
where ˘
q i is the expansion point in surface i. For analytical track models, the
Jacobian is also analytical. If, for example, the magnetic field is homogeneous,
the general case of propagation to a plane of arbitrary spatial orientation uses a
curvilinear coordinate frame moving along with the trajectory as a means of deriving
the required Jacobians [39].
In the general case of a non-analytical track model, the Jacobians cannot be
computed analytically either. The most straightforward approach is to calculate the
relevant derivatives in a purely numerical way. The basis for these calculations
is a reference trajectory corresponding to the expansion point. In addition, five
other trajectories are created, corresponding to small variations in each of the
track parameters. By propagating these five trajectories to the destination surface,
numerical derivatives can be obtained. A potential disadvantage of such an approach
is its computational complexity, as six trajectories have to be propagated instead of a
single one. Much less computational load is introduced by transporting the Jacobian
terms in parallel to the track parameters during the Runge–Kutta integration [40, 41],
avoiding the need for propagating auxiliary trajectories.
The measurement model describes the functional dependence of the measured
quantities on the state vector at a detector surface k:
m k = h k (q k ).
The vector of measurements m k usually contains the measured coordinates, but may
contain also other quantities, e.g. measurements of direction or even momentum. In
a pixel detector or in a double-sided silicon strip detector, m k is two-dimensional; in
a one-sided strip detector, it is one-dimensional. In a drift chamber or a multi-wire
proportional chamber with several layers, the measurement may be a track segment
resulting from an internal track reconstruction. In this case the vector m k may be
four- or five-dimensional, depending on whether the curvature can be estimated or
not.