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h2. This page describes the formulae used project tracking errors to the ACD |
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These are written up in greater detail |
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General framework
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(and with real formulae, instead of text-based approximations) here: [^formulae.pdf] \\ h4. General framework The tracker has a 4x4 representation {x_tkr} which uses a pair of slope-intercepts (x_0,y_0,m_x, m_y) to parametrize the position as a function of z. |
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This is great for a bunch of planes normal to the z axis, but really not so nice for planes normal to the x and y axes. |
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The first order of business is to transform the track parameterization to a 5x5 representation {x_acd} which uses a Point at Z=0 and a Vector (x_0,y_0, v_x,v_y,v_z) to parameterize the positions by arclength along the track (s). |
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We can the transform the covarience matrix by using the 5x4 track representation derivative matrix A = d x_acd / d x_tkr: |
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C_acd (5x5) = A C_tkr (4x4) A_T |
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If we then use the track parameterization to calculate a \[point of closest approach (POCA) or an intersection point {p}, the covarience on that point can be found using 5x3 solution derivative matrix B = d p / d A_acd with the covarience given by: C_p (3x3) = B C_acd (5x5) B_T |
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In the case where we care about the distance of closest approach to a point or a ray, we can project along the vector of closest approach J (the vector from the POCA to the point or ray in question). |
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Then we find |
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the error on the distance of closest approach: C_doca (1x1) = J C_p(3x3) J_T |
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In the case where we care about the intersection of a track with a plane, we can rotate into the plane using the rotation matrix R. |
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C_plane (3x3) = R C_p(3x3) R_T |
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and take only the upper-left 4 entries |
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Errors on POCA between a track and a point
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to get the 2x2 error ellipse on the surface of the plane. h4. Errors on POCA between a track and a point Call the point q, and the track (x,v) the POCA occurs at |
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The full derivative matrix is given in the attached write up.
Errors on POCA between a track and a ray
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an arclength s = (x - q) . v therefore the poca is p = x + sv and the vector of closest approach is: J = q-p = q - x + sv The full derivative matrix {B} is given in the attached write up. h4. Errors on POCA between a track and a ray Call the ray (r,q) and the track (x,v), the POCA occurs at an arclength along the track s = \[b(r.(x-q) - v.(x-q)\] / \[1-b*b\] and an arclength along the ray: t = \[r.(x-q) - b(v.(x-q))\] / \[1-b*b\] where b = v.r |
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Then the poca and vector of closest approach are: |
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p = x + sv J = (x-q) + sv - |
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The full derivative matrix is given in the attached write up.
Errors on point where a track crosses a plane
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tr The full derivative matrix {B} is given in the attached write up. h4. Errors on point where a track crosses a plane Given a plane defined by a point q and a rotation matrix R, the intersection occurs at: s = \[ (x-q).R_z \] / \[ v. R_z \] |
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Then the intersection point in the global frame is: |
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Of course we can get the intersection point in the local frame l:
l = R p + q
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p = x + sv
Of course we can get the intersection point in the local frame l:
l = R p + q
The full derivative matrix {B} is given in the attached write up.\\
h4. Some issues with the Error Projection
# They depend on the Kalman fitter hypothesis. Describing a 8GeV proton as a 100MeV electron (or vice-versa) is not going to give accurate errors
# They rely on knowing the material along the track path. For low energy particle the errors can blow up very quickly if there is a lot of material between the head of the track and the ACD
We distinguish between to ways of calculating the error projections:
# Full Propagation: track is kalman-propagated all the way back to the POCA and errors are calculated from the cov. matrix at the POCA
# Projection: errors are taken at the head of the track projected out at the POCA.
The errors from method 1) will alway be larger than the error from 2). For low-energy tracks the difference can be quite large.\\
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