Offline Data Analysis Software

Steve Snow, Paul S Miyagawa (University of Manchester)

John Hart (Rutherford Appleton Laboratory)

•Objectives + requirements•Field model + simulation•Pre-fit corrections•Fit results•Post-fit corrections•Conclusions

2 November 2005 2nd ATLAS Magnetic Field Workshop 2/12

Objectives• Momentum scale will be dominant uncertainty in W mass

measurement.• Momentum accuracy depends on ∫ r(rmax - r)Bzdr :

– Field at intermediate radii, as measured by the sagitta, is most important.

• In reality, the limit on silicon alignment under ideal conditions will be 1 μm.

• At typical sagitta of ~1 mm, we target an accuracy of 0.05% to ensure that B-field measurement is not the limiting factor on momentum accuracy.

• Field B may be described by a scalar potential satisfying Laplace’s equation, 2 = 0

• Sufficient to measure B on the surface of a cylinder (including the ends) surrounding the tracker

2 November 2005 2nd ATLAS Magnetic Field Workshop 3/12

Mapper Survey Requirements• Reinvestigated survey

requirements of mapper machine relative to Inner Detector.

• 1 mm survey error in x (or y) significant for high η tracks.

• 1 mm survey error in z significant for endcap tracks.

• 0.1 mrad rotation around x (or y) axis significant for high η tracks.

-8

-6

-4

-2

0

2

4

6

8

-2.5

-2.1

-1.7

-1.3

-0.9

-0.5

-0.1 0.

30.

71.

11.

51.

92.

3

Pseudorapidity

Re

lati

ve

sa

git

ta e

rro

r x

10

^4 phi 1

phi 2

phi 3

phi 4

phi 5

phi 6

phi 7

phi 8

phi 9

phi 10

Effect of 1 mm displacement in X direction

-5

-4

-3

-2

-1

0

1

2

3

4

5

-2.5

-2.1

-1.7

-1.3

-0.9

-0.5

-0.1 0.

30.

71.

11.

51.

92.

3

Pseudorapidity

Re

lati

ve

sa

git

ta e

rro

r x

10

^4 phi 1

phi 2

phi 3

phi 4

phi 5

phi 6

phi 7

phi 8

phi 9

phi 10

Effect of 1 mm displacement in Z direction

-8

-6

-4

-2

0

2

4

6

8

-2.5

-2.1

-1.7

-1.3

-0.9

-0.5

-0.1 0.

30.

71.

11.

51.

92.

3

Pseudorapidity

Re

lati

ve

sa

git

ta e

rro

r x

10

^4 phi 1

phi 2

phi 3

phi 4

phi 5

phi 6

phi 7

phi 8

phi 9

phi 10

Effect of 0.1 mrad rotation around the X axis

2 November 2005 2nd ATLAS Magnetic Field Workshop 4/12

Field Model

0

5000

10000

15000

20000

0

0.2

0.4

1

0.6

1

0.8

2

1.0

2

1.2

2

1.4

3

1.6

3

1.8

4

2.0

4

2.2

4

2.4

5

2.6

5

2.8

6

3.0

6

3.2

6

Z (m)

Fie

ld (

gau

ss)

r=0.2

r=0.4

r=0.6

r=0.8

r=1.0

r=1.2

r=0.2

r=0.4

r=0.6

r=0.8

r=1.0

r= 1.2

Z component

R component

• Basis of model is field due to coil of nominal dimensions.• Added in field due to magnetised iron (4% of total field). Distribution

of iron magnetisation taken from simple FEA model using FlexPDE.• Model is symmetric in and even in z.• Field can be displaced and rotated relative to mapping machine.

2 November 2005 2nd ATLAS Magnetic Field Workshop 5/12

Field Mapping Simulation• Field model sampled on a grid of

90 z-positions × 8 -angles (defined by encoder values of machine).

• 4 calibration points (2 near centre, 2 near one end) visited after every 25 measurements.

• Used current design of machine to determine positions of 48 Hall + 1 NMR probes.

• At each map point, wrote the following data to file:

– Time stamp– Solenoid current– Field modulus measured by 5

NMR probes (1 moving + 4 fixed)– 3 field components measured by

48 Hall probes

2 November 2005 2nd ATLAS Magnetic Field Workshop 6/12

Simulated ErrorsSix cumulative levels of error added to simulated data:1. No errors.2. Random errors in each measurement of:

• Solenoid current, 1 A• NMR B-field modulus, 0.1 G• Each component of Hall probe B-field, 3 G

3. Drifts by random walk process in each measurement of:• Solenoid current, 1 A• Each component of Hall probe B-field, 0.1 G

4. Random calibration scale and alignment errors, which are constant for each run:

• Each component of Hall probe B-field, 0.05% scale• Rotation about 3 axes of each Hall probe triplet, 0.1 mrad

5. Symmetry axis of field model displaced and rotated relative to the mapper machine axis. These misalignments are assumed to be measured perfectly by the surveyors.

6. Each Hall probe triplet has a systematic rotation of 1 mrad about the axis which mixes the Br and Bz components.

2 November 2005 2nd ATLAS Magnetic Field Workshop 7/12

Simulation of Raw Data• Field calculated from

solenoid of expected dimensions

• Magnetisation due to magnetic material from outside Inner Detector

• Random walk with time for solenoid current and Hall probe measurements

• Random errors for each measurement

2 November 2005 2nd ATLAS Magnetic Field Workshop 8/12

Correction for Current Drift• Average B-field of 4 NMR probes

used to calculate “actual” solenoid current.

• Scale all measurements to a reference current (7600 A).

• Effect of drift in current removed• Calibration capable of coping with

any sort of drift.

2 November 2005 2nd ATLAS Magnetic Field Workshop 9/12

Correction for Hall Probe Drift• Mapping machine regularly

returns to fixed calibration positions– Near coil centre to calibrate Bz

– Near coil end for Br

– No special calibration of B

• Each channel is calibrated to a reference time (beginning of run)

• Offsets from calibration points used to determine offsets for measurements between calibrations

2 November 2005 2nd ATLAS Magnetic Field Workshop 10/12

Geometrical Fit• Sum of simple fields known to obey Maxwell’s equations

– Long-thin coil (5 mm longer, 5 mm thinner than nominal)– Short-fat coil (5 mm shorter, 5 mm fatter)– Four terms of Fourier-Bessel series (for magnetisation)

• Use Minuit to minimise a 2 fit between the fitting function and simulated data using 12 free parameters:– 1, mixing ratio of long-thin and short-fat solenoids– 2, scaling factors for length and field of the combined solenoids– 4, coefficients of Fourier-Bessel series– 3, offset of field coordinates from mapper coordinates– 2, angles between field coordinates and mapper coordinates

• Fit dominated by large number of Hall probe points.– Could introduce one overall scale factor for all Hall probes as

free parameter and take true scale from NMR probes

2 November 2005 2nd ATLAS Magnetic Field Workshop 11/12

Fourier-Bessel Fit• General fit able to describe any field obeying Maxwell’s equations.• Uses only the field measurements on the surface of a bounding

cylinder, including the ends.• Parameterisation based on ALEPH 94-162 (Stephen Thorn) and

proceeds in three stages:

1. Bz on the cylindrical surface is fitted as Fourier series, giving terms with φ variation of form cos(nφ+α), with radial variation In(κr) (modified Bessel function).

2. Bzmeas – Bz(1) on the cylinder ends is fitted as a series of Bessel functions, Jn(λjr) where the λj are chosen so the terms vanish for r=rcyl. The z-dependence is of form cosh(μz) or sinh(μz).

3. The multipole terms are calculated from the measurements of Br on the cylindrical surface, averaged over z, after subtraction of the contribution to Br from the terms above. (The only relevant terms in Bz are those that are odd in z.)

2 November 2005 2nd ATLAS Magnetic Field Workshop 12/12

Fourier-Bessel FitComments:• The fit is very quick since the coefficients are calculated directly as

sums over the measured field except for a simple linear fit to the Jn coefficients.

• A typical fit could have up to ~500 terms in the first category taking into account combinations of 25 terms in z, 5 terms in φ and their respective phases, but most of the φ-dependent terms are too small to be measurable and are set to zero. In the second category, up to ~120 terms could be calculated but again most φ-dependent terms are negligible.

• The In and cosh or sinh terms decrease more or less exponentially with increasing distance from the outer cylinder or cylinder ends. This means that higher order terms contribute very little over most of the field volume and have little effect on the measured momentum.

• A poor fit indicates measurement errors rather than an incorrect model.

2 November 2005 2nd ATLAS Magnetic Field Workshop 13/12

Fit Quality with Ideal Performance• Quality of fit measured by comparing track sagitta in field model with track

sagitta in fitted field:– Used 260 track directions equally spaced in η and .

• Results shown below are for no errors added (error level 1):– Sagitta error not quite zero due to interpolation errors and finite number of terms.– Error is negligible compared to target level of 5×10-4.

-0.1

-0.05

0

0.05

0.1

0.15

-2.5 -1.5 -0.5 0.5 1.5 2.5

Pseudorapidity

Sa

git

ta e

rro

r x

10

^4

Geometrical

Fourier-Bessel

2 November 2005 2nd ATLAS Magnetic Field Workshop 14/12

Fit Quality with Expected Performance

• Expected mapper performance corresponds to error level 5.

• Both fits accurate within target level of 5×10-4.

0

0.1

0.2

0.3

0.4

0.5

0.6

Pseudorapidity

Re

lati

ve

sa

git

ta e

rro

r x

10

^4

phi 1

phi 2

phi 3

phi 4

phi 5

phi 6

phi 7

phi 8

phi 9

phi 10

Geometrical fit. Error level 5

-6

-4

-2

0

2

4

6

Pseudorapidity

Re

lati

ve

sa

git

ta e

rro

r x

10

^4

phi 1

phi 2

phi 3

phi 4

phi 5

phi 6

phi 7

phi 8

phi 9

phi 10

Fourier-Bessel fit. Error level 5

2 November 2005 2nd ATLAS Magnetic Field Workshop 15/12

Tables of Results

• Tables give statistics of Bz error over 25000 points and sagitta error over 260 trajectories.

• Correction procedure does not completely eliminate effects of drifts, so some runs (eg, 24) have normalisation errors of ~ 1×10-4.

• Geometrical fit relatively uninfluenced until error level 6 (systematic probe tilts).

Input file Bz error (G) Relative sagitta error x 10^4mean rms max min mean rms max min

CorrC21 0.00 0.74 3.24 -8.45 0.04 0.07 0.13 -0.07CorrC22 -0.65 1.04 4.00 -9.21 -0.32 0.33 -0.18 -0.48CorrC23 -0.33 0.83 2.89 -9.05 -0.13 0.14 0.01 -0.24CorrC24 -2.67 2.80 1.91 -9.44 -1.52 1.52 -1.34 -1.67CorrC25 0.40 1.06 4.06 -10.02 0.33 0.34 0.54 0.16CorrC26 -1.84 5.35 16.75 -24.62 -0.99 2.42 2.44 -4.62CorrC25 * 0.50 1.36 4.73 -9.28 0.37 0.56 1.13 -0.26CorrC26 * -1.79 2.08 3.02 -10.38 -0.97 1.04 -0.43 -1.70

Geometrical fitError level

Input file Bz error (G) Relative sagitta error x 10^4mean rms max min mean rms max min

CorrC21 -0.15 6.46 34.71 -43.83 -0.01 0.04 0.07 -0.07CorrC22 -1.16 6.96 38.15 -48.69 -0.56 1.08 1.25 -3.49CorrC23 -1.18 7.04 41.31 -48.06 -0.74 1.46 2.92 -4.71CorrC24 -2.16 13.01 64.57 -65.04 -1.05 4.26 5.13 -7.97CorrC25 0.70 9.15 48.99 -59.93 0.58 2.37 5.43 -4.95CorrC26 -5.43 11.79 39.52 -55.76 -3.32 4.43 5.59 -11.52CorrC25 * 1.66 7.14 44.19 -45.64 1.21 1.62 3.80 -1.37CorrC26 * -4.63 8.54 39.72 -49.42 -2.79 3.61 4.20 -10.70

Fourier-Bessel fit

* Fit results including probe normalisation and alignment corrections are shown with an asterisk by the file name.

2 November 2005 2nd ATLAS Magnetic Field Workshop 16/12

Probe Normalisation and Alignment Corrections

• Field should be very uniform near z=0, with Bz at maximum and Br=0. We take advantage of this information to perform two calibrations:

1. Inter-calibration of Bz scale between four Hall probes at common radius:

• Find Bz value at z=0 using quadratic fit in range |z| < 0.25 m.

• Normalise each probe to average scale of four at that radius.

2. Removal of z-r rotation:

1. MORE TEXT + PLOTS FOR THIS PAGE!

2 November 2005 2nd ATLAS Magnetic Field Workshop 17/12

Conclusions• Both fits have the necessary technical accuracy.• Both fits give results within target of 5×10-4.• Fourier-Bessel fit more sensitive to random

measurement errors:– Due to more free parameters.– F-B fit designed for any solenoid-like field, whereas

geometrical fit is specifically for the ATLAS solenoid.

• Probe normalisation + alignment correction helps correct for systematic tilts of Hall probes.

• With a few refinements, the code will be ready for release.