Kalman Filters for track fitting

(A.24)

A.2. Kalman Filters for track fitting

The application of the Kalman Filters to high energy particle tracking was first discussed in Frühwirth [45]. The Kalman Filter is computationally fast and flexible. If, in addition, the track model can be approximated sufficiently well by a discrete linear model in the neighborhood of the measurements belonging to an individual track, the Kalman Filter estimator has minimum variance among all linear estimators. Asymptotically, or in case of Gaussian measurement errors, this estimator is also efficient.

A.2.1. Linearization for tracks in a collimated beam

The discretized track model consists of a scattering at the current material layer and a vacuum propaga-tion to the next material layer. The scattering at layerkis given as

λ^?_k=g_ms(λ_k, η_k) (A.25)

while the vacuum propagation to the next layerk+ 1is written as

λk+1 =fk+1|k(λ^?_k) (A.26)

Mathematically, the track model can be seen as a compositionf_k+1|k•g_kof the scattering function and the vacuum track propagator.

λ_k+1=f_k+1|k(g_ms(λ_k, η_k)) (A.27)

Following Frühwirth [66], the idea is to linearize the track model around a reference trajectory which is a solution of the vacuum propagator and stays close to the true particle trajectory. In our case, a natural choice for a reference trajectory is the axis of the collimated particle beam. In the following, the vector λk,rdenotes the parameters of the beam axis at telescope layerk. To first order, the deviations between the parameters of a real particleλ_kand the parameters of the beam axis can be propagated linearly.

λ_k+1−λ_k+1,r =F_k+1|k(λ_k−λ_k,r) +F_k+1|kG_kw_k (A.28) Here,F_k+1|kis the transport matrix defined as the derivative to the vacuum propagator along the beam axis.

F_k+1|k= ∂f_k+1|k

∂λ_{k |λk=λk,r} (A.29)

Similarly, the scatter matrixG_kis defined as the derivative of the scatter function at pointλ_k,r. G_k = ∂g_ms

∂λk |λ=λ_k,r

(A.30)

The matrices F_k+1|k and G_k were computed in the previous section. We can introduce linearized parameter vectorsxkand linearized measurementsmk,r in the following way:

m_k,r = m_k−Hλ_k,r (A.31)

x_k = λ_k−λ_k,r (A.32)

To complete the formal definition of a linearized dynamical system for Kalman filtering, the so called system equation for the parametersx_kcan be formulated as

xk+1=Fk+1|kxk+Fk+1|kGkwk (A.33)

while the measurement equation is

m_k,r =Hx_k+_k. (A.34)

The Kalman Filter provides estimates for the track parameter differences x_k and their covariance matrixC_k. The predicted track parameter differencesx^p_kand their covariance matrixC_k^p are computed using all measurementm_j,ron upstream telescope layersj < k. The indexpindicates that the estimate is a prediction ahead in time using previous measurements on the position of the particle. Similarly, the updated parameter estimatesx^up_k andC_k^upare computed using all previous measurement including the measurement at layerk. The recursive system of equations needed to compute predicted and updated estimates for the linear dynamical system defined in Eq. A.36 and A.34 is described in detail in the literature; see for example [78].

The Kalman Filter also requires an initial estimatex^p₀andC₀^pof the track parameter difference before the first measurement is taken. The method developed for this thesis is to employ the collimated beam model for filter initialization. The first layer with indexk = 0is thez = 0plane and we usex^p₀ = 0 andC₀^p = C_b (as given in Eq. A.2) to initialize the Kalman Filter. An alternative way to initialize the Kalman Filter without putting much emphasize on the beam parameters is to useC₀^p=αCbwith a large scaling factorα. For this alternative initialization case, square root filters offer the advantage to allow an initial covarianceC₀^pfor the limiting caseα→ ∞.

A.2.2. Forward, backward and time reversed filters

In the context of track fitting, the above described filter for the track parametersx_k is called aforward filter. The dynamic model defined in Eq. A.36 follows the particle from layerk = 0to layerk =nin the flight direction. The scattering of the particle at the sensor layers implies that a distinction between local parameters before and after scattering at any given layer is necessary. In particular, the forward filter estimates the parametersλkof the particle as it enters a sensor layer before scattering takes place.

To complement the forward filter, we can define a backward filter in the following way. The linear mapping relating the particle state at layerk+ 1to the state at layerkis given as:

x_k=F_k+1|k⁻¹ x_k+1−G_kw_k. (A.35)

Eq. A.35 can be used as the system equation for the backward filter, that allows to process

mea-surements in the reverse order starting with the measurement at layerk =n. Especially, the backward filter computes predicted parameter estimatesx^b,p_k using all measurements at layerj > k and updated estimatesx^b,up_k using all measurementsj=k.

There is another possibility to run the dynamic system given in Eq. A.33 in the opposite direction, namely time reversal. The system equation for the time reversed dynamic is given as

x^?_k =F_k|k+1x^?_k+1+F_k|k+1G_k+1w_k+1. (A.36) Obviously, the time reversed system must be formulated for the state of the particlex^?_kafter scattering at the material in layerk. Using the notion developed in the previous section, we can write this in the following way:

x^?_k=λ^?_k−λ_k,r. (A.37)

The time reversed system can be used to process measurement in the reverse order starting with the measurement at layer k = n and processing against the flight direction of the particle. Unlike the backward filter, the time reversed filter estimates the local particle state after scattering at layerk. In this thesis, the main use of the time reversed filter is to estimate the scattering kinksη_kfrom differences∆_k between the prediction of the forward filterx^p_kand the prediction of the time reversed filterx^?,p_k .

∆_k=x^?,p_k −x^p_k (A.38)

As the forward filter uses all measurements at sensor before layerkand the time reversed filter uses all measurements behind, it is obvious thatx^?,p_k andx^p_kare uncorrelated. As a consequence, the covariance for the difference∆_kis simply given by

C_∆k =C_k^?,p+C_k^p. (A.39)

A.2.3. Track fitting with forward and backward filters

By running a forward and a backward filter at the same time, we can produce precise estimators for track parameters at layerkusing hits before and after layerkat the same time. To simplify the notation, we will drop the uppercase indexpand denote the predicted estimators from the forward and backward fil-ters asx^f_k, C_k^f andx^b_k, C_k^b. By construction, the these estimators are uncorrelated for any layerkbecause they use non-overlapping sets of measurements. Therefore, we can compute a smoothed estimator for the local parameters at layerkas a weighted means.

Ck =

C_k^f −1

+

C_k^b

−1−1

(A.40) x_k=C_k

C_k^f−1

x^f_k+ C_k^b−1

x^b_k

(A.41) The smoothed parameter vectorx_kand its covarianceC_kcombine the information from all measure-ments excluding the measurement at layer kitself. In other words,xk andmk are uncorrelated. The smoothed track parameter vector λ_k at thekth sensor is obtained as the sum x_k+λ_k,r and has the

track p-value 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

tracks

0 200 400 600 800 1000

Figure A.1.: Distribution of trackp-values for simulated tracks from100randomly tilted telescope se-tups in a1GeV electron beam.

covariance matrixCkdefined in Eq. A.40. For track fitting in the EUDET telescope, the fitting method is implemented in a way to compute smoothed local track parameters for all sensors in the telescope.