13 Pattern Recognition and Reconstruction
567
If the direction is represented by the direction cosines c x = dx/ds and c y = dy/ds,
the covariance matrix is [45]
Var[((c x , ,c y )
T
] = σ
2
P
(1 − c x ) 2 −c x c y
−c x c y (1 − c y ) 2
.
In all cases the projected variance σ 2
P takes into account the effective amount of
material crossed by the track.
If the transverse offset cannot be neglected, its variance and its correlation with
the angle have to be taken into account. Assume that the particle passes a scatterer of
length d, traveling along the z-axis. Neglecting the curvature of the track, the joint
covariance matrix of the offset x and the scattering angle θ x in the x–z projection
is
var((x) = σ
2
0 d
3 /3, var(θ x ) = σ
2
0 d, cov((x, θ x ) = σ
2
0 d
2 /2,
where σ 2
0 is the variance of the projected scattering angle per unit length. If the
particle enters the scatterer at z = 0 with direction (t x , t y ), the joint covariance
matrix of the offsets x, ,y and the angles x , ,t y at z = d is
Var
⎡
⎢
⎢
⎢
⎢
⎣
⎛
⎜
⎜
⎜
⎜
⎝
x
y
x
y
⎞
⎟
⎟
⎟
⎟
⎠
⎤
⎥
⎥
⎥
⎥
⎦
= σ
2
0 (1+t
2
x +t
2
y )
⎛
⎜
⎜
⎜
⎜
⎝
(1 + t 2
x )D 3 /3 t x t y D 3 /3 (1 + t 2
x )D 2 /2 t x t y D 2 /2
t x t y D 3 /3 (1 + t 2
y )D 3 /3 t x t y D 2 /2 (1 + t 2
y )D 2 /2
(1 + t 2
x )D 2 /2 t x t y D 2 /2
(1 + t 2
x )D
t x t y D
t x t y D 2 /2 (1 + t 2
y )D 2 /2
t x t y D
( 1 + t 2
y )D
⎞
⎟
⎟
⎟
⎟
⎠
,
where D = (1 + t 2
x + t 2
y ) 1/2 d is the effective thickness crossed. If the direction
is represented by θ and ϕ, the covariance matrix can be computed via the
transformation
t x = tan(θ ) cos(ϕ), t y = tan(θ ) sin(ϕ),
and linear error propagation with the Jacobian
T =
∂(θ, ϕ)
∂(t x , t y )
=
⎛
⎝
t x
t 2
x +t 2
y (1+t 2
x +t 2
y )
t y
t 2
x +t 2
y (1+t 2
x +t 2
y )
−
t y
t 2
x +t 2
y
t x
t 2
x +t 2
y
⎞
⎠ =
cos(ϕ)
1+tan 2 (θ)
sin(ϕ)
1+tan 2 (θ)
−
sin(ϕ)
tan(θ)
cos(ϕ)
tan(θ)
.
For analogous formulas in cylindrical coordinates, see [46].
If the curvature of the track cannot be neglected, the simplest approach is
a stepwise integration of the equation of motion, assuming the validity of a
helical track model within each step and considering each such step as a thin
scatterer [39, 47].